TPTP Problem File: SLH0413^1.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_00099_003389__14662288_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1447 ( 662 unt; 165 typ; 0 def)
% Number of atoms : 3552 (1291 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 10733 ( 268 ~; 64 |; 210 &;8840 @)
% ( 0 <=>;1351 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 587 ( 587 >; 0 *; 0 +; 0 <<)
% Number of symbols : 158 ( 155 usr; 19 con; 0-3 aty)
% Number of variables : 3348 ( 243 ^;3065 !; 40 ?;3348 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:43:31.368
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_Mt__Extended____Nonnegative____Real__Oennreal_J_J,type,
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thf(ty_n_t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
set_Ex3793607809372303086nnreal: $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__Sigma____Algebra__Omeasure_Itf__a_J_J,type,
set_Sigma_measure_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__Extended____Nonnegative____Real__Oennreal,type,
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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_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (155)
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001tf__a,type,
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thf(sy_c_Caratheodory_Olambda__system_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Caratheodory_Olambda__system_001tf__a,type,
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thf(sy_c_Caratheodory_Ovolume_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Caratheodory_Ovolume_001tf__a,type,
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thf(sy_c_Complete__Measure_Ocld__measure_001tf__a,type,
comple4590207612277428097sure_a: sigma_measure_a > $o ).
thf(sy_c_Complete__Measure_Ocomplete__measure_001tf__a,type,
comple8155536527497655953sure_a: sigma_measure_a > $o ).
thf(sy_c_Complete__Measure_Olocally__determined__measure_001tf__a,type,
comple7257097362500340559sure_a: sigma_measure_a > $o ).
thf(sy_c_Complete__Measure_Olocally__determined__measure__axioms_001tf__a,type,
comple7282333422040910444ioms_a: sigma_measure_a > $o ).
thf(sy_c_Complete__Measure_Omeasurable__envelope_001tf__a,type,
comple1317578422686860819lope_a: sigma_measure_a > set_a > set_a > $o ).
thf(sy_c_Complete__Measure_Onull__part_001tf__a,type,
complete_null_part_a: sigma_measure_a > set_a > set_a ).
thf(sy_c_Complete__Measure_Oouter__measure__of_001tf__a,type,
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thf(sy_c_Complete__Measure_Osemifinite__measure_001tf__a,type,
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thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_If_001tf__a,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
sup_sup_set_a_o: ( set_a > $o ) > ( set_a > $o ) > set_a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_Itf__a_J_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
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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,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
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thf(sy_c_Measure__Space_Ofinite__measure_001tf__a,type,
measur930452917991658466sure_a: sigma_measure_a > $o ).
thf(sy_c_Measure__Space_Ofmeasurable_001t__Set__Oset_Itf__a_J,type,
measur7460903245211743562_set_a: sigma_measure_set_a > set_set_set_a ).
thf(sy_c_Measure__Space_Ofmeasurable_001tf__a,type,
measur3645360004775918570able_a: sigma_measure_a > set_set_a ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001tf__a,type,
measur632008712001695539ng_a_a: set_set_a > ( set_a > a ) > $o ).
thf(sy_c_Measure__Space_Onull__sets_001t__Set__Oset_Itf__a_J,type,
measur1516554128032400785_set_a: sigma_measure_set_a > set_set_set_a ).
thf(sy_c_Measure__Space_Onull__sets_001tf__a,type,
measure_null_sets_a: sigma_measure_a > set_set_a ).
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__Extended____Nonnegative____Real__Oennreal,type,
down_r5963364644869667083nnreal: set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_Median_Odown__ray_001tf__a,type,
down_ray_a: set_a > $o ).
thf(sy_c_Median_Ointerval_001t__Extended____Nonnegative____Real__Oennreal,type,
interv3853198846103503604nnreal: set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Median_Oup__ray_001t__Extended____Nonnegative____Real__Oennreal,type,
up_ray4546996785294415186nnreal: set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_Median_Oup__ray_001tf__a,type,
up_ray_a: set_a > $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_It__Set__Oset_Itf__a_J_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo3380559777022489994_set_a: set_set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Sigma____Algebra__Omeasure_Itf__a_J_J,type,
bot_bo5171090233048072389sure_a: set_Sigma_measure_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_Itf__a_J,type,
bot_bo2108912051383640591sure_a: sigma_measure_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le7381754540660121996nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $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_M_Eo_J,type,
ord_less_eq_set_a_o: ( set_a > $o ) > ( set_a > $o ) > $o ).
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ord_le6700572704167691815nnreal: ( set_a > extend8495563244428889912nnreal ) > ( set_a > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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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_It__Sigma____Algebra__Omeasure_Itf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_Itf__a_J,type,
ord_max_set_a: set_a > set_a > set_a ).
thf(sy_c_Orderings_Oord__class_Omax_001tf__a,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Omin_001tf__a,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
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top_top_set_a: set_a ).
thf(sy_c_Radon__Nikodym_Odiff__measure_001tf__a,type,
radon_diff_measure_a: sigma_measure_a > sigma_measure_a > sigma_measure_a ).
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,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Set_Oinsert_001tf__a,type,
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thf(sy_c_Set_Ois__empty_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Set_Ois__empty_001tf__a,type,
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thf(sy_c_Set_Ois__singleton_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Set_Ois__singleton_001tf__a,type,
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thf(sy_c_Set_Oremove_001tf__a,type,
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thf(sy_c_Set_Othe__elem_001tf__a,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_Itf__a_J,type,
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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__Extended____Nonnegative____Real__Oennreal,type,
set_or8062060312790847405nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal ).
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_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
set_or6558828460305013104sure_a: sigma_measure_a > set_Sigma_measure_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__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_001t__Extended____Nonnegative____Real__Oennreal,type,
set_or4252868341878644913nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal ).
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_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
set_or6272550623556448372sure_a: sigma_measure_a > set_Sigma_measure_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__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__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_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_Itf__a_J,type,
set_or460448635090783044_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001tf__a,type,
set_or8632414552788122084Than_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_Itf__a_J,type,
set_or5421148953861284865_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001tf__a,type,
set_ord_lessThan_a: a > set_a ).
thf(sy_c_Sigma__Algebra_Oalgebra_001t__Set__Oset_Itf__a_J,type,
sigma_algebra_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Oalgebra_001tf__a,type,
sigma_algebra_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Oclosed__cdi_001t__Set__Oset_Itf__a_J,type,
sigma_8605711284820113968_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Oclosed__cdi_001tf__a,type,
sigma_closed_cdi_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Oemeasure_001t__Set__Oset_Itf__a_J,type,
sigma_emeasure_set_a: sigma_measure_set_a > set_set_a > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001tf__a,type,
sigma_emeasure_a: sigma_measure_a > set_a > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oring__of__sets_001t__Set__Oset_Itf__a_J,type,
sigma_5899762507328327638_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Oring__of__sets_001tf__a,type,
sigma_ring_of_sets_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osemiring__of__sets_001t__Set__Oset_Itf__a_J,type,
sigma_6809455824285593704_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osemiring__of__sets_001tf__a,type,
sigma_8461971822185508616sets_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osemiring__of__sets_Ogenerated__ring_001t__Set__Oset_Itf__a_J,type,
sigma_7997604194779675523_set_a: set_set_set_a > set_set_set_a ).
thf(sy_c_Sigma__Algebra_Osemiring__of__sets_Ogenerated__ring_001tf__a,type,
sigma_6990475660951004707ring_a: set_set_a > set_set_a ).
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_Osigma__algebra_001tf__a,type,
sigma_4968961713055010667ebra_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osmallest__ccdi__sets_001t__Set__Oset_Itf__a_J,type,
sigma_8136683933974138297_set_a: set_set_a > set_set_set_a > set_set_set_a ).
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__Set__Oset_Itf__a_J,type,
sigma_space_set_a: sigma_measure_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__a,type,
sigma_space_a: sigma_measure_a > set_a ).
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__Extended____Nonnegative____Real__Oennreal,type,
member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $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_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
member3534519376729797778sure_a: sigma_measure_a > set_Sigma_measure_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (1278)
thf(fact_0_down__ray__def,axiom,
( down_r5963364644869667083nnreal
= ( ^ [I: set_Ex3793607809372303086nnreal] :
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Y @ I )
=> ( ( ord_le3935885782089961368nnreal @ X @ Y )
=> ( member7908768830364227535nnreal @ X @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_1_down__ray__def,axiom,
( down_ray_a
= ( ^ [I: set_a] :
! [X: a,Y: a] :
( ( member_a @ Y @ I )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( member_a @ X @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_2_assms,axiom,
interval_a @ i ).
% assms
thf(fact_3_a,axiom,
member_a @ x @ i ).
% a
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_AI_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X2: a] :
~ ( member_a @ X2 @ i ) ).
% \<open>\<And>thesis. (\<And>x. x \<in> I \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5__092_060open_062_092_060And_062z_Ay_O_A_092_060lbrakk_062y_A_092_060in_062_AI_A_092_060union_062_A_123_O_Ox_125_059_Az_A_092_060le_062_Ay_092_060rbrakk_062_A_092_060Longrightarrow_062_Az_A_092_060in_062_AI_A_092_060union_062_A_123_O_Ox_125_092_060close_062,axiom,
! [Y2: a,Z: a] :
( ( member_a @ Y2 @ ( sup_sup_set_a @ i @ ( set_ord_atMost_a @ x ) ) )
=> ( ( ord_less_eq_a @ Z @ Y2 )
=> ( member_a @ Z @ ( sup_sup_set_a @ i @ ( set_ord_atMost_a @ x ) ) ) ) ) ).
% \<open>\<And>z y. \<lbrakk>y \<in> I \<union> {..x}; z \<le> y\<rbrakk> \<Longrightarrow> z \<in> I \<union> {..x}\<close>
thf(fact_6_False,axiom,
i != bot_bot_set_a ).
% False
thf(fact_7__092_060open_062_092_060And_062z_Ay_O_A_092_060lbrakk_062y_A_092_060in_062_AI_A_092_060union_062_A_123x_O_O_125_059_Ay_A_092_060le_062_Az_092_060rbrakk_062_A_092_060Longrightarrow_062_Az_A_092_060in_062_AI_A_092_060union_062_A_123x_O_O_125_092_060close_062,axiom,
! [Y2: a,Z: a] :
( ( member_a @ Y2 @ ( sup_sup_set_a @ i @ ( set_ord_atLeast_a @ x ) ) )
=> ( ( ord_less_eq_a @ Y2 @ Z )
=> ( member_a @ Z @ ( sup_sup_set_a @ i @ ( set_ord_atLeast_a @ x ) ) ) ) ) ).
% \<open>\<And>z y. \<lbrakk>y \<in> I \<union> {x..}; y \<le> z\<rbrakk> \<Longrightarrow> z \<in> I \<union> {x..}\<close>
thf(fact_8_atMost__eq__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( ( set_ord_atMost_set_a @ X3 )
= ( set_ord_atMost_set_a @ Y2 ) )
= ( X3 = Y2 ) ) ).
% atMost_eq_iff
thf(fact_9_atMost__eq__iff,axiom,
! [X3: a,Y2: a] :
( ( ( set_ord_atMost_a @ X3 )
= ( set_ord_atMost_a @ Y2 ) )
= ( X3 = Y2 ) ) ).
% atMost_eq_iff
thf(fact_10_UnCI,axiom,
! [C: set_set_a,B: set_set_set_a,A: set_set_set_a] :
( ( ~ ( member_set_set_a @ C @ B )
=> ( member_set_set_a @ C @ A ) )
=> ( member_set_set_a @ C @ ( sup_su2076012971530813682_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_11_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_12_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_13_Un__iff,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( sup_su2076012971530813682_set_a @ A @ B ) )
= ( ( member_set_set_a @ C @ A )
| ( member_set_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_14_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_15_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_16_sup_Oidem,axiom,
! [A2: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_17_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_18_sup_Oidem,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_19_sup__idem,axiom,
! [X3: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_20_sup__idem,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_21_sup__idem,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_22_sup_Oleft__idem,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ A2 @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) )
= ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_23_sup_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_24_sup_Oleft__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_25_sup__left__idem,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ X3 @ ( sup_su27664952386392231sure_a @ X3 @ Y2 ) )
= ( sup_su27664952386392231sure_a @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_26_sup__left__idem,axiom,
! [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
= ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_27_sup__left__idem,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ X3 @ Y2 ) )
= ( sup_sup_set_set_a @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_28_sup_Oright__idem,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) @ B2 )
= ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_29_sup_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_30_sup_Oright__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_31__092_060open_062up__ray_A_II_A_092_060union_062_A_123x_O_O_125_J_092_060close_062,axiom,
up_ray_a @ ( sup_sup_set_a @ i @ ( set_ord_atLeast_a @ x ) ) ).
% \<open>up_ray (I \<union> {x..})\<close>
thf(fact_32_UnE,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( sup_su2076012971530813682_set_a @ A @ B ) )
=> ( ~ ( member_set_set_a @ C @ A )
=> ( member_set_set_a @ C @ B ) ) ) ).
% UnE
thf(fact_33_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_34_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_35_empty__iff,axiom,
! [C: set_set_a] :
~ ( member_set_set_a @ C @ bot_bo3380559777022489994_set_a ) ).
% empty_iff
thf(fact_36_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_37_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_38_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_39_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_40_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_41_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_42_all__not__in__conv,axiom,
! [A: set_set_set_a] :
( ( ! [X: set_set_a] :
~ ( member_set_set_a @ X @ A ) )
= ( A = bot_bo3380559777022489994_set_a ) ) ).
% all_not_in_conv
thf(fact_43_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X: set_a] :
~ ( member_set_a @ X @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_44_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_45_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_46_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_47_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_48_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_49_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_50_Un__subset__iff,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 )
= ( ( ord_le3724670747650509150_set_a @ A @ C2 )
& ( ord_le3724670747650509150_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_51_atLeast__eq__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( ( set_or8362275514725411625_set_a @ X3 )
= ( set_or8362275514725411625_set_a @ Y2 ) )
= ( X3 = Y2 ) ) ).
% atLeast_eq_iff
thf(fact_52_atLeast__eq__iff,axiom,
! [X3: a,Y2: a] :
( ( ( set_ord_atLeast_a @ X3 )
= ( set_ord_atLeast_a @ Y2 ) )
= ( X3 = Y2 ) ) ).
% atLeast_eq_iff
thf(fact_53_sup_Obounded__iff,axiom,
! [B2: sigma_measure_a,C: sigma_measure_a,A2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ ( sup_su27664952386392231sure_a @ B2 @ C ) @ A2 )
= ( ( ord_le254669795585780187sure_a @ B2 @ A2 )
& ( ord_le254669795585780187sure_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_54_sup_Obounded__iff,axiom,
! [B2: set_a > extend8495563244428889912nnreal,C: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ ( sup_su1017758682940751451nnreal @ B2 @ C ) @ A2 )
= ( ( ord_le6700572704167691815nnreal @ B2 @ A2 )
& ( ord_le6700572704167691815nnreal @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_55_sup_Obounded__iff,axiom,
! [B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ B2 @ C ) @ A2 )
= ( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
& ( ord_le3935885782089961368nnreal @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_56_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_57_sup_Obounded__iff,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 )
= ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
& ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_58_le__sup__iff,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a,Z: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ ( sup_su27664952386392231sure_a @ X3 @ Y2 ) @ Z )
= ( ( ord_le254669795585780187sure_a @ X3 @ Z )
& ( ord_le254669795585780187sure_a @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_59_le__sup__iff,axiom,
! [X3: set_a > extend8495563244428889912nnreal,Y2: set_a > extend8495563244428889912nnreal,Z: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ ( sup_su1017758682940751451nnreal @ X3 @ Y2 ) @ Z )
= ( ( ord_le6700572704167691815nnreal @ X3 @ Z )
& ( ord_le6700572704167691815nnreal @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_60_le__sup__iff,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ X3 @ Y2 ) @ Z )
= ( ( ord_le3935885782089961368nnreal @ X3 @ Z )
& ( ord_le3935885782089961368nnreal @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_61_le__sup__iff,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
= ( ( ord_less_eq_set_a @ X3 @ Z )
& ( ord_less_eq_set_a @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_62_le__sup__iff,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X3 @ Y2 ) @ Z )
= ( ( ord_le3724670747650509150_set_a @ X3 @ Z )
& ( ord_le3724670747650509150_set_a @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_63_sup__bot_Oright__neutral,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ A2 @ bot_bot_a_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_64_sup__bot_Oright__neutral,axiom,
! [A2: set_a > $o] :
( ( sup_sup_set_a_o @ A2 @ bot_bot_set_a_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_65_sup__bot_Oright__neutral,axiom,
! [A2: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ A2 @ bot_bo2108912051383640591sure_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_66_sup__bot_Oright__neutral,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_67_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_68_sup__bot_Oneutr__eq__iff,axiom,
! [A2: a > $o,B2: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_a_o )
& ( B2 = bot_bot_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_69_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a > $o,B2: set_a > $o] :
( ( bot_bot_set_a_o
= ( sup_sup_set_a_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_a_o )
& ( B2 = bot_bot_set_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_70_sup__bot_Oneutr__eq__iff,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a] :
( ( bot_bo2108912051383640591sure_a
= ( sup_su27664952386392231sure_a @ A2 @ B2 ) )
= ( ( A2 = bot_bo2108912051383640591sure_a )
& ( B2 = bot_bo2108912051383640591sure_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_71_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_72_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_73_sup__bot_Oleft__neutral,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_74_sup__bot_Oleft__neutral,axiom,
! [A2: set_a > $o] :
( ( sup_sup_set_a_o @ bot_bot_set_a_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_75_sup__bot_Oleft__neutral,axiom,
! [A2: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ bot_bo2108912051383640591sure_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_76_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_77_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_78_sup__bot_Oeq__neutr__iff,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ( sup_sup_a_o @ A2 @ B2 )
= bot_bot_a_o )
= ( ( A2 = bot_bot_a_o )
& ( B2 = bot_bot_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_79_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a > $o,B2: set_a > $o] :
( ( ( sup_sup_set_a_o @ A2 @ B2 )
= bot_bot_set_a_o )
= ( ( A2 = bot_bot_set_a_o )
& ( B2 = bot_bot_set_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_80_sup__bot_Oeq__neutr__iff,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a] :
( ( ( sup_su27664952386392231sure_a @ A2 @ B2 )
= bot_bo2108912051383640591sure_a )
= ( ( A2 = bot_bo2108912051383640591sure_a )
& ( B2 = bot_bo2108912051383640591sure_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_81_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_82_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_83_sup__eq__bot__iff,axiom,
! [X3: a > $o,Y2: a > $o] :
( ( ( sup_sup_a_o @ X3 @ Y2 )
= bot_bot_a_o )
= ( ( X3 = bot_bot_a_o )
& ( Y2 = bot_bot_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_84_sup__eq__bot__iff,axiom,
! [X3: set_a > $o,Y2: set_a > $o] :
( ( ( sup_sup_set_a_o @ X3 @ Y2 )
= bot_bot_set_a_o )
= ( ( X3 = bot_bot_set_a_o )
& ( Y2 = bot_bot_set_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_85_sup__eq__bot__iff,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a] :
( ( ( sup_su27664952386392231sure_a @ X3 @ Y2 )
= bot_bo2108912051383640591sure_a )
= ( ( X3 = bot_bo2108912051383640591sure_a )
& ( Y2 = bot_bo2108912051383640591sure_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_86_sup__eq__bot__iff,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( ( sup_sup_set_set_a @ X3 @ Y2 )
= bot_bot_set_set_a )
= ( ( X3 = bot_bot_set_set_a )
& ( Y2 = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_87_sup__eq__bot__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( ( sup_sup_set_a @ X3 @ Y2 )
= bot_bot_set_a )
= ( ( X3 = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_88_bot__eq__sup__iff,axiom,
! [X3: a > $o,Y2: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_a_o )
& ( Y2 = bot_bot_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_89_bot__eq__sup__iff,axiom,
! [X3: set_a > $o,Y2: set_a > $o] :
( ( bot_bot_set_a_o
= ( sup_sup_set_a_o @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_set_a_o )
& ( Y2 = bot_bot_set_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_90_bot__eq__sup__iff,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a] :
( ( bot_bo2108912051383640591sure_a
= ( sup_su27664952386392231sure_a @ X3 @ Y2 ) )
= ( ( X3 = bot_bo2108912051383640591sure_a )
& ( Y2 = bot_bo2108912051383640591sure_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_91_bot__eq__sup__iff,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_set_set_a )
& ( Y2 = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_92_bot__eq__sup__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_93_sup__bot__right,axiom,
! [X3: a > $o] :
( ( sup_sup_a_o @ X3 @ bot_bot_a_o )
= X3 ) ).
% sup_bot_right
thf(fact_94_sup__bot__right,axiom,
! [X3: set_a > $o] :
( ( sup_sup_set_a_o @ X3 @ bot_bot_set_a_o )
= X3 ) ).
% sup_bot_right
thf(fact_95_sup__bot__right,axiom,
! [X3: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ X3 @ bot_bo2108912051383640591sure_a )
= X3 ) ).
% sup_bot_right
thf(fact_96_sup__bot__right,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ bot_bot_set_set_a )
= X3 ) ).
% sup_bot_right
thf(fact_97_sup__bot__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ bot_bot_set_a )
= X3 ) ).
% sup_bot_right
thf(fact_98_sup__bot__left,axiom,
! [X3: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_99_sup__bot__left,axiom,
! [X3: set_a > $o] :
( ( sup_sup_set_a_o @ bot_bot_set_a_o @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_100_sup__bot__left,axiom,
! [X3: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ bot_bo2108912051383640591sure_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_101_sup__bot__left,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_102_sup__bot__left,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_103_Un__empty,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_104_Un__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_105_atMost__subset__iff,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4016371710855203973_set_a @ X3 ) @ ( set_or4016371710855203973_set_a @ Y2 ) )
= ( ord_le3724670747650509150_set_a @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_106_atMost__subset__iff,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a] :
( ( ord_le2601950661898901649sure_a @ ( set_or6272550623556448372sure_a @ X3 ) @ ( set_or6272550623556448372sure_a @ Y2 ) )
= ( ord_le254669795585780187sure_a @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_107_atMost__subset__iff,axiom,
! [X3: set_a > extend8495563244428889912nnreal,Y2: set_a > extend8495563244428889912nnreal] :
( ( ord_le2718372912061172615nnreal @ ( set_or7674582181447345870nnreal @ X3 ) @ ( set_or7674582181447345870nnreal @ Y2 ) )
= ( ord_le6700572704167691815nnreal @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_108_atMost__subset__iff,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ ( set_or4252868341878644913nnreal @ X3 ) @ ( set_or4252868341878644913nnreal @ Y2 ) )
= ( ord_le3935885782089961368nnreal @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_109_atMost__subset__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X3 ) @ ( set_ord_atMost_set_a @ Y2 ) )
= ( ord_less_eq_set_a @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_110_atMost__subset__iff,axiom,
! [X3: a,Y2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ X3 ) @ ( set_ord_atMost_a @ Y2 ) )
= ( ord_less_eq_a @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_111_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_112_atMost__iff,axiom,
! [I2: sigma_measure_a,K: sigma_measure_a] :
( ( member3534519376729797778sure_a @ I2 @ ( set_or6272550623556448372sure_a @ K ) )
= ( ord_le254669795585780187sure_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_113_atMost__iff,axiom,
! [I2: set_a > extend8495563244428889912nnreal,K: set_a > extend8495563244428889912nnreal] :
( ( member4180043592386426928nnreal @ I2 @ ( set_or7674582181447345870nnreal @ K ) )
= ( ord_le6700572704167691815nnreal @ I2 @ K ) ) ).
% atMost_iff
thf(fact_114_atMost__iff,axiom,
! [I2: extend8495563244428889912nnreal,K: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I2 @ ( set_or4252868341878644913nnreal @ K ) )
= ( ord_le3935885782089961368nnreal @ I2 @ K ) ) ).
% atMost_iff
thf(fact_115_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_116_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_117_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_118_atLeast__iff,axiom,
! [I2: sigma_measure_a,K: sigma_measure_a] :
( ( member3534519376729797778sure_a @ I2 @ ( set_or6558828460305013104sure_a @ K ) )
= ( ord_le254669795585780187sure_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_119_atLeast__iff,axiom,
! [I2: set_a > extend8495563244428889912nnreal,K: set_a > extend8495563244428889912nnreal] :
( ( member4180043592386426928nnreal @ I2 @ ( set_or3682978270108489938nnreal @ K ) )
= ( ord_le6700572704167691815nnreal @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_120_atLeast__iff,axiom,
! [I2: extend8495563244428889912nnreal,K: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I2 @ ( set_or8062060312790847405nnreal @ K ) )
= ( ord_le3935885782089961368nnreal @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_121_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_122_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_123_atLeast__subset__iff,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or3904034815786525833_set_a @ X3 ) @ ( set_or3904034815786525833_set_a @ Y2 ) )
= ( ord_le3724670747650509150_set_a @ Y2 @ X3 ) ) ).
% atLeast_subset_iff
thf(fact_124_atLeast__subset__iff,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a] :
( ( ord_le2601950661898901649sure_a @ ( set_or6558828460305013104sure_a @ X3 ) @ ( set_or6558828460305013104sure_a @ Y2 ) )
= ( ord_le254669795585780187sure_a @ Y2 @ X3 ) ) ).
% atLeast_subset_iff
thf(fact_125_atLeast__subset__iff,axiom,
! [X3: set_a > extend8495563244428889912nnreal,Y2: set_a > extend8495563244428889912nnreal] :
( ( ord_le2718372912061172615nnreal @ ( set_or3682978270108489938nnreal @ X3 ) @ ( set_or3682978270108489938nnreal @ Y2 ) )
= ( ord_le6700572704167691815nnreal @ Y2 @ X3 ) ) ).
% atLeast_subset_iff
thf(fact_126_atLeast__subset__iff,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ ( set_or8062060312790847405nnreal @ X3 ) @ ( set_or8062060312790847405nnreal @ Y2 ) )
= ( ord_le3935885782089961368nnreal @ Y2 @ X3 ) ) ).
% atLeast_subset_iff
thf(fact_127_atLeast__subset__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or8362275514725411625_set_a @ X3 ) @ ( set_or8362275514725411625_set_a @ Y2 ) )
= ( ord_less_eq_set_a @ Y2 @ X3 ) ) ).
% atLeast_subset_iff
thf(fact_128_atLeast__subset__iff,axiom,
! [X3: a,Y2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ X3 ) @ ( set_ord_atLeast_a @ Y2 ) )
= ( ord_less_eq_a @ Y2 @ X3 ) ) ).
% atLeast_subset_iff
thf(fact_129_not__empty__eq__Ici__eq__empty,axiom,
! [L: set_a] :
( bot_bot_set_set_a
!= ( set_or8362275514725411625_set_a @ L ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_130_not__empty__eq__Ici__eq__empty,axiom,
! [L: a] :
( bot_bot_set_a
!= ( set_ord_atLeast_a @ L ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_131_emptyE,axiom,
! [A2: set_set_a] :
~ ( member_set_set_a @ A2 @ bot_bo3380559777022489994_set_a ) ).
% emptyE
thf(fact_132_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_133_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_134_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_135_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_136_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_137_Collect__mem__eq,axiom,
! [A: set_set_set_a] :
( ( collect_set_set_a
@ ^ [X: set_set_a] : ( member_set_set_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_138_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_140_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_141_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_142_equals0D,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( A = bot_bo3380559777022489994_set_a )
=> ~ ( member_set_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_143_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_144_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_145_equals0I,axiom,
! [A: set_set_set_a] :
( ! [Y3: set_set_a] :
~ ( member_set_set_a @ Y3 @ A )
=> ( A = bot_bo3380559777022489994_set_a ) ) ).
% equals0I
thf(fact_146_equals0I,axiom,
! [A: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_147_equals0I,axiom,
! [A: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_148_ex__in__conv,axiom,
! [A: set_set_set_a] :
( ( ? [X: set_set_a] : ( member_set_set_a @ X @ A ) )
= ( A != bot_bo3380559777022489994_set_a ) ) ).
% ex_in_conv
thf(fact_149_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X: set_a] : ( member_set_a @ X @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_150_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_151_up__ray__def,axiom,
( up_ray4546996785294415186nnreal
= ( ^ [I: set_Ex3793607809372303086nnreal] :
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ I )
=> ( ( ord_le3935885782089961368nnreal @ X @ Y )
=> ( member7908768830364227535nnreal @ Y @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_152_up__ray__def,axiom,
( up_ray_a
= ( ^ [I: set_a] :
! [X: a,Y: a] :
( ( member_a @ X @ I )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( member_a @ Y @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_153_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_154_subset__Un__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_155_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ A )
=> ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ B )
=> ( C2
!= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_156_subset__UnE,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) )
=> ~ ! [A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ A )
=> ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ B )
=> ( C2
!= ( sup_sup_set_set_a @ A4 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_157_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_158_Un__absorb2,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( sup_sup_set_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_159_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_160_Un__absorb1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( sup_sup_set_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_161_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_162_Un__upper2,axiom,
! [B: set_set_a,A: set_set_a] : ( ord_le3724670747650509150_set_a @ B @ ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_163_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_164_Un__upper1,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_165_Un__least,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_166_Un__least,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_167_Un__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_168_Un__mono,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_169_Un__empty__right,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Un_empty_right
thf(fact_170_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_171_Un__empty__left,axiom,
! [B: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_172_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_173_not__empty__eq__Iic__eq__empty,axiom,
! [H: set_a] :
( bot_bot_set_set_a
!= ( set_ord_atMost_set_a @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_174_not__empty__eq__Iic__eq__empty,axiom,
! [H: a] :
( bot_bot_set_a
!= ( set_ord_atMost_a @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_175_interval__def,axiom,
( interv3853198846103503604nnreal
= ( ^ [I: set_Ex3793607809372303086nnreal] :
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ I )
=> ( ( member7908768830364227535nnreal @ Z2 @ I )
=> ( ( ord_le3935885782089961368nnreal @ X @ Y )
=> ( ( ord_le3935885782089961368nnreal @ Y @ Z2 )
=> ( member7908768830364227535nnreal @ Y @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_176_interval__def,axiom,
( interval_a
= ( ^ [I: set_a] :
! [X: a,Y: a,Z2: a] :
( ( member_a @ X @ I )
=> ( ( member_a @ Z2 @ I )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z2 )
=> ( member_a @ Y @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_177_sup_OcoboundedI2,axiom,
! [C: sigma_measure_a,B2: sigma_measure_a,A2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ C @ B2 )
=> ( ord_le254669795585780187sure_a @ C @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_178_sup_OcoboundedI2,axiom,
! [C: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ C @ B2 )
=> ( ord_le6700572704167691815nnreal @ C @ ( sup_su1017758682940751451nnreal @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_179_sup_OcoboundedI2,axiom,
! [C: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C @ B2 )
=> ( ord_le3935885782089961368nnreal @ C @ ( sup_su6922871097908479076nnreal @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_180_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_181_sup_OcoboundedI2,axiom,
! [C: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_182_sup_OcoboundedI1,axiom,
! [C: sigma_measure_a,A2: sigma_measure_a,B2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ C @ A2 )
=> ( ord_le254669795585780187sure_a @ C @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_183_sup_OcoboundedI1,axiom,
! [C: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ C @ A2 )
=> ( ord_le6700572704167691815nnreal @ C @ ( sup_su1017758682940751451nnreal @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_184_sup_OcoboundedI1,axiom,
! [C: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C @ A2 )
=> ( ord_le3935885782089961368nnreal @ C @ ( sup_su6922871097908479076nnreal @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_185_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_186_sup_OcoboundedI1,axiom,
! [C: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_187_sup_Oabsorb__iff2,axiom,
( ord_le254669795585780187sure_a
= ( ^ [A5: sigma_measure_a,B5: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_188_sup_Oabsorb__iff2,axiom,
( ord_le6700572704167691815nnreal
= ( ^ [A5: set_a > extend8495563244428889912nnreal,B5: set_a > extend8495563244428889912nnreal] :
( ( sup_su1017758682940751451nnreal @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_189_sup_Oabsorb__iff2,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( sup_su6922871097908479076nnreal @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_190_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_191_sup_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( sup_sup_set_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_192_sup_Oabsorb__iff1,axiom,
( ord_le254669795585780187sure_a
= ( ^ [B5: sigma_measure_a,A5: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_193_sup_Oabsorb__iff1,axiom,
( ord_le6700572704167691815nnreal
= ( ^ [B5: set_a > extend8495563244428889912nnreal,A5: set_a > extend8495563244428889912nnreal] :
( ( sup_su1017758682940751451nnreal @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_194_sup_Oabsorb__iff1,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [B5: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal] :
( ( sup_su6922871097908479076nnreal @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_195_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_196_sup_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B5: set_set_a,A5: set_set_a] :
( ( sup_sup_set_set_a @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_197_sup_Ocobounded2,axiom,
! [B2: sigma_measure_a,A2: sigma_measure_a] : ( ord_le254669795585780187sure_a @ B2 @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_198_sup_Ocobounded2,axiom,
! [B2: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ B2 @ ( sup_su1017758682940751451nnreal @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_199_sup_Ocobounded2,axiom,
! [B2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ B2 @ ( sup_su6922871097908479076nnreal @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_200_sup_Ocobounded2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_201_sup_Ocobounded2,axiom,
! [B2: set_set_a,A2: set_set_a] : ( ord_le3724670747650509150_set_a @ B2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_202_sup_Ocobounded1,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a] : ( ord_le254669795585780187sure_a @ A2 @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_203_sup_Ocobounded1,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ A2 @ ( sup_su1017758682940751451nnreal @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_204_sup_Ocobounded1,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A2 @ ( sup_su6922871097908479076nnreal @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_205_sup_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_206_sup_Ocobounded1,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_207_sup_Oorder__iff,axiom,
( ord_le254669795585780187sure_a
= ( ^ [B5: sigma_measure_a,A5: sigma_measure_a] :
( A5
= ( sup_su27664952386392231sure_a @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_208_sup_Oorder__iff,axiom,
( ord_le6700572704167691815nnreal
= ( ^ [B5: set_a > extend8495563244428889912nnreal,A5: set_a > extend8495563244428889912nnreal] :
( A5
= ( sup_su1017758682940751451nnreal @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_209_sup_Oorder__iff,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [B5: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal] :
( A5
= ( sup_su6922871097908479076nnreal @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_210_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A5: set_a] :
( A5
= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_211_sup_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B5: set_set_a,A5: set_set_a] :
( A5
= ( sup_sup_set_set_a @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_212_sup_OboundedI,axiom,
! [B2: sigma_measure_a,A2: sigma_measure_a,C: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ B2 @ A2 )
=> ( ( ord_le254669795585780187sure_a @ C @ A2 )
=> ( ord_le254669795585780187sure_a @ ( sup_su27664952386392231sure_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_213_sup_OboundedI,axiom,
! [B2: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal,C: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ B2 @ A2 )
=> ( ( ord_le6700572704167691815nnreal @ C @ A2 )
=> ( ord_le6700572704167691815nnreal @ ( sup_su1017758682940751451nnreal @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_214_sup_OboundedI,axiom,
! [B2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ C @ A2 )
=> ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_215_sup_OboundedI,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_216_sup_OboundedI,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_217_sup_OboundedE,axiom,
! [B2: sigma_measure_a,C: sigma_measure_a,A2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ ( sup_su27664952386392231sure_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le254669795585780187sure_a @ B2 @ A2 )
=> ~ ( ord_le254669795585780187sure_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_218_sup_OboundedE,axiom,
! [B2: set_a > extend8495563244428889912nnreal,C: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ ( sup_su1017758682940751451nnreal @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le6700572704167691815nnreal @ B2 @ A2 )
=> ~ ( ord_le6700572704167691815nnreal @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_219_sup_OboundedE,axiom,
! [B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
=> ~ ( ord_le3935885782089961368nnreal @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_220_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_221_sup_OboundedE,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ~ ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_222_sup__absorb2,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X3 @ Y2 )
=> ( ( sup_su27664952386392231sure_a @ X3 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_223_sup__absorb2,axiom,
! [X3: set_a > extend8495563244428889912nnreal,Y2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ X3 @ Y2 )
=> ( ( sup_su1017758682940751451nnreal @ X3 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_224_sup__absorb2,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ( sup_su6922871097908479076nnreal @ X3 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_225_sup__absorb2,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( sup_sup_set_a @ X3 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_226_sup__absorb2,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ( sup_sup_set_set_a @ X3 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_227_sup__absorb1,axiom,
! [Y2: sigma_measure_a,X3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ Y2 @ X3 )
=> ( ( sup_su27664952386392231sure_a @ X3 @ Y2 )
= X3 ) ) ).
% sup_absorb1
thf(fact_228_sup__absorb1,axiom,
! [Y2: set_a > extend8495563244428889912nnreal,X3: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ Y2 @ X3 )
=> ( ( sup_su1017758682940751451nnreal @ X3 @ Y2 )
= X3 ) ) ).
% sup_absorb1
thf(fact_229_sup__absorb1,axiom,
! [Y2: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y2 @ X3 )
=> ( ( sup_su6922871097908479076nnreal @ X3 @ Y2 )
= X3 ) ) ).
% sup_absorb1
thf(fact_230_sup__absorb1,axiom,
! [Y2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( ( sup_sup_set_a @ X3 @ Y2 )
= X3 ) ) ).
% sup_absorb1
thf(fact_231_sup__absorb1,axiom,
! [Y2: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X3 )
=> ( ( sup_sup_set_set_a @ X3 @ Y2 )
= X3 ) ) ).
% sup_absorb1
thf(fact_232_sup_Oabsorb2,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ B2 )
=> ( ( sup_su27664952386392231sure_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_233_sup_Oabsorb2,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A2 @ B2 )
=> ( ( sup_su1017758682940751451nnreal @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_234_sup_Oabsorb2,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
=> ( ( sup_su6922871097908479076nnreal @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_235_sup_Oabsorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_236_sup_Oabsorb2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_237_sup_Oabsorb1,axiom,
! [B2: sigma_measure_a,A2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ B2 @ A2 )
=> ( ( sup_su27664952386392231sure_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_238_sup_Oabsorb1,axiom,
! [B2: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ B2 @ A2 )
=> ( ( sup_su1017758682940751451nnreal @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_239_sup_Oabsorb1,axiom,
! [B2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
=> ( ( sup_su6922871097908479076nnreal @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_240_sup_Oabsorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_241_sup_Oabsorb1,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_242_sup__unique,axiom,
! [F: sigma_measure_a > sigma_measure_a > sigma_measure_a,X3: sigma_measure_a,Y2: sigma_measure_a] :
( ! [X2: sigma_measure_a,Y3: sigma_measure_a] : ( ord_le254669795585780187sure_a @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: sigma_measure_a,Y3: sigma_measure_a] : ( ord_le254669795585780187sure_a @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: sigma_measure_a,Y3: sigma_measure_a,Z3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ Y3 @ X2 )
=> ( ( ord_le254669795585780187sure_a @ Z3 @ X2 )
=> ( ord_le254669795585780187sure_a @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su27664952386392231sure_a @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_243_sup__unique,axiom,
! [F: ( set_a > extend8495563244428889912nnreal ) > ( set_a > extend8495563244428889912nnreal ) > set_a > extend8495563244428889912nnreal,X3: set_a > extend8495563244428889912nnreal,Y2: set_a > extend8495563244428889912nnreal] :
( ! [X2: set_a > extend8495563244428889912nnreal,Y3: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a > extend8495563244428889912nnreal,Y3: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a > extend8495563244428889912nnreal,Y3: set_a > extend8495563244428889912nnreal,Z3: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ Y3 @ X2 )
=> ( ( ord_le6700572704167691815nnreal @ Z3 @ X2 )
=> ( ord_le6700572704167691815nnreal @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su1017758682940751451nnreal @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_244_sup__unique,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ( ( ord_le3935885782089961368nnreal @ Z3 @ X2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su6922871097908479076nnreal @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_245_sup__unique,axiom,
! [F: set_a > set_a > set_a,X3: set_a,Y2: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_246_sup__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X3: set_set_a,Y2: set_set_a] :
( ! [X2: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ Z3 @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_set_a @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_247_sup_OorderI,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a] :
( ( A2
= ( sup_su27664952386392231sure_a @ A2 @ B2 ) )
=> ( ord_le254669795585780187sure_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_248_sup_OorderI,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal] :
( ( A2
= ( sup_su1017758682940751451nnreal @ A2 @ B2 ) )
=> ( ord_le6700572704167691815nnreal @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_249_sup_OorderI,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( A2
= ( sup_su6922871097908479076nnreal @ A2 @ B2 ) )
=> ( ord_le3935885782089961368nnreal @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_250_sup_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_251_sup_OorderI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2
= ( sup_sup_set_set_a @ A2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_252_sup_OorderE,axiom,
! [B2: sigma_measure_a,A2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ B2 @ A2 )
=> ( A2
= ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_253_sup_OorderE,axiom,
! [B2: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ B2 @ A2 )
=> ( A2
= ( sup_su1017758682940751451nnreal @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_254_sup_OorderE,axiom,
! [B2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
=> ( A2
= ( sup_su6922871097908479076nnreal @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_255_sup_OorderE,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_256_sup_OorderE,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_257_le__iff__sup,axiom,
( ord_le254669795585780187sure_a
= ( ^ [X: sigma_measure_a,Y: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_258_le__iff__sup,axiom,
( ord_le6700572704167691815nnreal
= ( ^ [X: set_a > extend8495563244428889912nnreal,Y: set_a > extend8495563244428889912nnreal] :
( ( sup_su1017758682940751451nnreal @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_259_le__iff__sup,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( sup_su6922871097908479076nnreal @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_260_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_261_le__iff__sup,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_262_sup__least,axiom,
! [Y2: sigma_measure_a,X3: sigma_measure_a,Z: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ Y2 @ X3 )
=> ( ( ord_le254669795585780187sure_a @ Z @ X3 )
=> ( ord_le254669795585780187sure_a @ ( sup_su27664952386392231sure_a @ Y2 @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_263_sup__least,axiom,
! [Y2: set_a > extend8495563244428889912nnreal,X3: set_a > extend8495563244428889912nnreal,Z: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ Y2 @ X3 )
=> ( ( ord_le6700572704167691815nnreal @ Z @ X3 )
=> ( ord_le6700572704167691815nnreal @ ( sup_su1017758682940751451nnreal @ Y2 @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_264_sup__least,axiom,
! [Y2: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y2 @ X3 )
=> ( ( ord_le3935885782089961368nnreal @ Z @ X3 )
=> ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ Y2 @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_265_sup__least,axiom,
! [Y2: set_a,X3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( ( ord_less_eq_set_a @ Z @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y2 @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_266_sup__least,axiom,
! [Y2: set_set_a,X3: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X3 )
=> ( ( ord_le3724670747650509150_set_a @ Z @ X3 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ Y2 @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_267_sup__mono,axiom,
! [A2: sigma_measure_a,C: sigma_measure_a,B2: sigma_measure_a,D2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ C )
=> ( ( ord_le254669795585780187sure_a @ B2 @ D2 )
=> ( ord_le254669795585780187sure_a @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) @ ( sup_su27664952386392231sure_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_268_sup__mono,axiom,
! [A2: set_a > extend8495563244428889912nnreal,C: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal,D2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A2 @ C )
=> ( ( ord_le6700572704167691815nnreal @ B2 @ D2 )
=> ( ord_le6700572704167691815nnreal @ ( sup_su1017758682940751451nnreal @ A2 @ B2 ) @ ( sup_su1017758682940751451nnreal @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_269_sup__mono,axiom,
! [A2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ C )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ D2 )
=> ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ A2 @ B2 ) @ ( sup_su6922871097908479076nnreal @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_270_sup__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_271_sup__mono,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( sup_sup_set_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_272_sup_Omono,axiom,
! [C: sigma_measure_a,A2: sigma_measure_a,D2: sigma_measure_a,B2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ C @ A2 )
=> ( ( ord_le254669795585780187sure_a @ D2 @ B2 )
=> ( ord_le254669795585780187sure_a @ ( sup_su27664952386392231sure_a @ C @ D2 ) @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_273_sup_Omono,axiom,
! [C: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal,D2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ C @ A2 )
=> ( ( ord_le6700572704167691815nnreal @ D2 @ B2 )
=> ( ord_le6700572704167691815nnreal @ ( sup_su1017758682940751451nnreal @ C @ D2 ) @ ( sup_su1017758682940751451nnreal @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_274_sup_Omono,axiom,
! [C: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ D2 @ B2 )
=> ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ C @ D2 ) @ ( sup_su6922871097908479076nnreal @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_275_sup_Omono,axiom,
! [C: set_a,A2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_276_sup_Omono,axiom,
! [C: set_set_a,A2: set_set_a,D2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ D2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ C @ D2 ) @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_277_le__supI2,axiom,
! [X3: sigma_measure_a,B2: sigma_measure_a,A2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X3 @ B2 )
=> ( ord_le254669795585780187sure_a @ X3 @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_278_le__supI2,axiom,
! [X3: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ X3 @ B2 )
=> ( ord_le6700572704167691815nnreal @ X3 @ ( sup_su1017758682940751451nnreal @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_279_le__supI2,axiom,
! [X3: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ B2 )
=> ( ord_le3935885782089961368nnreal @ X3 @ ( sup_su6922871097908479076nnreal @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_280_le__supI2,axiom,
! [X3: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X3 @ B2 )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_281_le__supI2,axiom,
! [X3: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ B2 )
=> ( ord_le3724670747650509150_set_a @ X3 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_282_le__supI1,axiom,
! [X3: sigma_measure_a,A2: sigma_measure_a,B2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X3 @ A2 )
=> ( ord_le254669795585780187sure_a @ X3 @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_283_le__supI1,axiom,
! [X3: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ X3 @ A2 )
=> ( ord_le6700572704167691815nnreal @ X3 @ ( sup_su1017758682940751451nnreal @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_284_le__supI1,axiom,
! [X3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ A2 )
=> ( ord_le3935885782089961368nnreal @ X3 @ ( sup_su6922871097908479076nnreal @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_285_le__supI1,axiom,
! [X3: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_286_le__supI1,axiom,
! [X3: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X3 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_287_sup__ge2,axiom,
! [Y2: sigma_measure_a,X3: sigma_measure_a] : ( ord_le254669795585780187sure_a @ Y2 @ ( sup_su27664952386392231sure_a @ X3 @ Y2 ) ) ).
% sup_ge2
thf(fact_288_sup__ge2,axiom,
! [Y2: set_a > extend8495563244428889912nnreal,X3: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ Y2 @ ( sup_su1017758682940751451nnreal @ X3 @ Y2 ) ) ).
% sup_ge2
thf(fact_289_sup__ge2,axiom,
! [Y2: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ ( sup_su6922871097908479076nnreal @ X3 @ Y2 ) ) ).
% sup_ge2
thf(fact_290_sup__ge2,axiom,
! [Y2: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% sup_ge2
thf(fact_291_sup__ge2,axiom,
! [Y2: set_set_a,X3: set_set_a] : ( ord_le3724670747650509150_set_a @ Y2 @ ( sup_sup_set_set_a @ X3 @ Y2 ) ) ).
% sup_ge2
thf(fact_292_sup__ge1,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a] : ( ord_le254669795585780187sure_a @ X3 @ ( sup_su27664952386392231sure_a @ X3 @ Y2 ) ) ).
% sup_ge1
thf(fact_293_sup__ge1,axiom,
! [X3: set_a > extend8495563244428889912nnreal,Y2: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ X3 @ ( sup_su1017758682940751451nnreal @ X3 @ Y2 ) ) ).
% sup_ge1
thf(fact_294_sup__ge1,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X3 @ ( sup_su6922871097908479076nnreal @ X3 @ Y2 ) ) ).
% sup_ge1
thf(fact_295_sup__ge1,axiom,
! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% sup_ge1
thf(fact_296_sup__ge1,axiom,
! [X3: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ X3 @ ( sup_sup_set_set_a @ X3 @ Y2 ) ) ).
% sup_ge1
thf(fact_297_le__supI,axiom,
! [A2: sigma_measure_a,X3: sigma_measure_a,B2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ X3 )
=> ( ( ord_le254669795585780187sure_a @ B2 @ X3 )
=> ( ord_le254669795585780187sure_a @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_298_le__supI,axiom,
! [A2: set_a > extend8495563244428889912nnreal,X3: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A2 @ X3 )
=> ( ( ord_le6700572704167691815nnreal @ B2 @ X3 )
=> ( ord_le6700572704167691815nnreal @ ( sup_su1017758682940751451nnreal @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_299_le__supI,axiom,
! [A2: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ X3 )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ X3 )
=> ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_300_le__supI,axiom,
! [A2: set_a,X3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X3 )
=> ( ( ord_less_eq_set_a @ B2 @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_301_le__supI,axiom,
! [A2: set_set_a,X3: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ X3 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ X3 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_302_le__supE,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a,X3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le254669795585780187sure_a @ A2 @ X3 )
=> ~ ( ord_le254669795585780187sure_a @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_303_le__supE,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal,X3: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ ( sup_su1017758682940751451nnreal @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le6700572704167691815nnreal @ A2 @ X3 )
=> ~ ( ord_le6700572704167691815nnreal @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_304_le__supE,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le3935885782089961368nnreal @ A2 @ X3 )
=> ~ ( ord_le3935885782089961368nnreal @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_305_le__supE,axiom,
! [A2: set_a,B2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X3 )
=> ~ ( ord_less_eq_set_a @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_306_le__supE,axiom,
! [A2: set_set_a,B2: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ X3 )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_307_inf__sup__ord_I3_J,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a] : ( ord_le254669795585780187sure_a @ X3 @ ( sup_su27664952386392231sure_a @ X3 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_308_inf__sup__ord_I3_J,axiom,
! [X3: set_a > extend8495563244428889912nnreal,Y2: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ X3 @ ( sup_su1017758682940751451nnreal @ X3 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_309_inf__sup__ord_I3_J,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X3 @ ( sup_su6922871097908479076nnreal @ X3 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_310_inf__sup__ord_I3_J,axiom,
! [X3: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_311_inf__sup__ord_I3_J,axiom,
! [X3: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ X3 @ ( sup_sup_set_set_a @ X3 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_312_inf__sup__ord_I4_J,axiom,
! [Y2: sigma_measure_a,X3: sigma_measure_a] : ( ord_le254669795585780187sure_a @ Y2 @ ( sup_su27664952386392231sure_a @ X3 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_313_inf__sup__ord_I4_J,axiom,
! [Y2: set_a > extend8495563244428889912nnreal,X3: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ Y2 @ ( sup_su1017758682940751451nnreal @ X3 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_314_inf__sup__ord_I4_J,axiom,
! [Y2: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ ( sup_su6922871097908479076nnreal @ X3 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_315_inf__sup__ord_I4_J,axiom,
! [Y2: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_316_inf__sup__ord_I4_J,axiom,
! [Y2: set_set_a,X3: set_set_a] : ( ord_le3724670747650509150_set_a @ Y2 @ ( sup_sup_set_set_a @ X3 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_317_sup__left__commute,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a,Z: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ X3 @ ( sup_su27664952386392231sure_a @ Y2 @ Z ) )
= ( sup_su27664952386392231sure_a @ Y2 @ ( sup_su27664952386392231sure_a @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_318_sup__left__commute,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_319_sup__left__commute,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y2 @ Z ) )
= ( sup_sup_set_set_a @ Y2 @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_320_sup_Oleft__commute,axiom,
! [B2: sigma_measure_a,A2: sigma_measure_a,C: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ B2 @ ( sup_su27664952386392231sure_a @ A2 @ C ) )
= ( sup_su27664952386392231sure_a @ A2 @ ( sup_su27664952386392231sure_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_321_sup_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_322_sup_Oleft__commute,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ B2 @ ( sup_sup_set_set_a @ A2 @ C ) )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_323_sup__commute,axiom,
( sup_su27664952386392231sure_a
= ( ^ [X: sigma_measure_a,Y: sigma_measure_a] : ( sup_su27664952386392231sure_a @ Y @ X ) ) ) ).
% sup_commute
thf(fact_324_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X ) ) ) ).
% sup_commute
thf(fact_325_sup__commute,axiom,
( sup_sup_set_set_a
= ( ^ [X: set_set_a,Y: set_set_a] : ( sup_sup_set_set_a @ Y @ X ) ) ) ).
% sup_commute
thf(fact_326_sup_Ocommute,axiom,
( sup_su27664952386392231sure_a
= ( ^ [A5: sigma_measure_a,B5: sigma_measure_a] : ( sup_su27664952386392231sure_a @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_327_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_328_sup_Ocommute,axiom,
( sup_sup_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] : ( sup_sup_set_set_a @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_329_sup__assoc,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a,Z: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ ( sup_su27664952386392231sure_a @ X3 @ Y2 ) @ Z )
= ( sup_su27664952386392231sure_a @ X3 @ ( sup_su27664952386392231sure_a @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_330_sup__assoc,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_331_sup__assoc,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_332_sup_Oassoc,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a,C: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) @ C )
= ( sup_su27664952386392231sure_a @ A2 @ ( sup_su27664952386392231sure_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_333_sup_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_334_sup_Oassoc,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ C )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_335_inf__sup__aci_I5_J,axiom,
( sup_su27664952386392231sure_a
= ( ^ [X: sigma_measure_a,Y: sigma_measure_a] : ( sup_su27664952386392231sure_a @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_336_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_337_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_a
= ( ^ [X: set_set_a,Y: set_set_a] : ( sup_sup_set_set_a @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_338_inf__sup__aci_I6_J,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a,Z: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ ( sup_su27664952386392231sure_a @ X3 @ Y2 ) @ Z )
= ( sup_su27664952386392231sure_a @ X3 @ ( sup_su27664952386392231sure_a @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_339_inf__sup__aci_I6_J,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_340_inf__sup__aci_I6_J,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_341_inf__sup__aci_I7_J,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a,Z: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ X3 @ ( sup_su27664952386392231sure_a @ Y2 @ Z ) )
= ( sup_su27664952386392231sure_a @ Y2 @ ( sup_su27664952386392231sure_a @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_342_inf__sup__aci_I7_J,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_343_inf__sup__aci_I7_J,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y2 @ Z ) )
= ( sup_sup_set_set_a @ Y2 @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_344_inf__sup__aci_I8_J,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a] :
( ( sup_su27664952386392231sure_a @ X3 @ ( sup_su27664952386392231sure_a @ X3 @ Y2 ) )
= ( sup_su27664952386392231sure_a @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_345_inf__sup__aci_I8_J,axiom,
! [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
= ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_346_inf__sup__aci_I8_J,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ X3 @ Y2 ) )
= ( sup_sup_set_set_a @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_347_Un__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_348_Un__left__commute,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) )
= ( sup_sup_set_set_a @ B @ ( sup_sup_set_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_349_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_350_Un__left__absorb,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ A @ B ) )
= ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_351_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_352_Un__commute,axiom,
( sup_sup_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] : ( sup_sup_set_set_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_353_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_354_Un__absorb,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_355_Un__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_356_Un__assoc,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_357_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X: a] :
( ( member_a @ X @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( P @ X ) )
& ! [X: a] :
( ( member_a @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_358_ball__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ! [X: set_a] :
( ( member_set_a @ X @ ( sup_sup_set_set_a @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( P @ X ) )
& ! [X: set_a] :
( ( member_set_a @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_359_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X: a] :
( ( member_a @ X @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: a] :
( ( member_a @ X @ A )
& ( P @ X ) )
| ? [X: a] :
( ( member_a @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_360_bex__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ? [X: set_a] :
( ( member_set_a @ X @ ( sup_sup_set_set_a @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( P @ X ) )
| ? [X: set_a] :
( ( member_set_a @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_361_UnI2,axiom,
! [C: set_set_a,B: set_set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ C @ B )
=> ( member_set_set_a @ C @ ( sup_su2076012971530813682_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_362_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_363_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_364_UnI1,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ A )
=> ( member_set_set_a @ C @ ( sup_su2076012971530813682_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_365_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_366_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_367_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_368_bot__apply,axiom,
( bot_bot_set_a_o
= ( ^ [X: set_a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_369_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_370_order__refl,axiom,
! [X3: set_set_a] : ( ord_le3724670747650509150_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_371_order__refl,axiom,
! [X3: sigma_measure_a] : ( ord_le254669795585780187sure_a @ X3 @ X3 ) ).
% order_refl
thf(fact_372_order__refl,axiom,
! [X3: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ X3 @ X3 ) ).
% order_refl
thf(fact_373_order__refl,axiom,
! [X3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X3 @ X3 ) ).
% order_refl
thf(fact_374_order__refl,axiom,
! [X3: a] : ( ord_less_eq_a @ X3 @ X3 ) ).
% order_refl
thf(fact_375_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_376_dual__order_Orefl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_377_dual__order_Orefl,axiom,
! [A2: sigma_measure_a] : ( ord_le254669795585780187sure_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_378_dual__order_Orefl,axiom,
! [A2: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_379_dual__order_Orefl,axiom,
! [A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_380_dual__order_Orefl,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_381_boolean__algebra_Odisj__zero__right,axiom,
! [X3: a > $o] :
( ( sup_sup_a_o @ X3 @ bot_bot_a_o )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_382_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_a > $o] :
( ( sup_sup_set_a_o @ X3 @ bot_bot_set_a_o )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_383_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ bot_bot_set_set_a )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_384_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ bot_bot_set_a )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_385_bot_Oextremum,axiom,
! [A2: a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A2 ) ).
% bot.extremum
thf(fact_386_bot_Oextremum,axiom,
! [A2: set_a > $o] : ( ord_less_eq_set_a_o @ bot_bot_set_a_o @ A2 ) ).
% bot.extremum
thf(fact_387_bot_Oextremum,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% bot.extremum
thf(fact_388_bot_Oextremum,axiom,
! [A2: sigma_measure_a] : ( ord_le254669795585780187sure_a @ bot_bo2108912051383640591sure_a @ A2 ) ).
% bot.extremum
thf(fact_389_bot_Oextremum,axiom,
! [A2: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ bot_bo1361193569059450611nnreal @ A2 ) ).
% bot.extremum
thf(fact_390_bot_Oextremum,axiom,
! [A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ bot_bo841427958541957580nnreal @ A2 ) ).
% bot.extremum
thf(fact_391_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_392_bot_Oextremum__unique,axiom,
! [A2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ bot_bot_a_o )
= ( A2 = bot_bot_a_o ) ) ).
% bot.extremum_unique
thf(fact_393_bot_Oextremum__unique,axiom,
! [A2: set_a > $o] :
( ( ord_less_eq_set_a_o @ A2 @ bot_bot_set_a_o )
= ( A2 = bot_bot_set_a_o ) ) ).
% bot.extremum_unique
thf(fact_394_bot_Oextremum__unique,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_395_bot_Oextremum__unique,axiom,
! [A2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ bot_bo2108912051383640591sure_a )
= ( A2 = bot_bo2108912051383640591sure_a ) ) ).
% bot.extremum_unique
thf(fact_396_bot_Oextremum__unique,axiom,
! [A2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A2 @ bot_bo1361193569059450611nnreal )
= ( A2 = bot_bo1361193569059450611nnreal ) ) ).
% bot.extremum_unique
thf(fact_397_bot_Oextremum__unique,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ bot_bo841427958541957580nnreal )
= ( A2 = bot_bo841427958541957580nnreal ) ) ).
% bot.extremum_unique
thf(fact_398_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_399_bot_Oextremum__uniqueI,axiom,
! [A2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ bot_bot_a_o )
=> ( A2 = bot_bot_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_400_bot_Oextremum__uniqueI,axiom,
! [A2: set_a > $o] :
( ( ord_less_eq_set_a_o @ A2 @ bot_bot_set_a_o )
=> ( A2 = bot_bot_set_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_401_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
=> ( A2 = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_402_bot_Oextremum__uniqueI,axiom,
! [A2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ bot_bo2108912051383640591sure_a )
=> ( A2 = bot_bo2108912051383640591sure_a ) ) ).
% bot.extremum_uniqueI
thf(fact_403_bot_Oextremum__uniqueI,axiom,
! [A2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A2 @ bot_bo1361193569059450611nnreal )
=> ( A2 = bot_bo1361193569059450611nnreal ) ) ).
% bot.extremum_uniqueI
thf(fact_404_bot_Oextremum__uniqueI,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ bot_bo841427958541957580nnreal )
=> ( A2 = bot_bo841427958541957580nnreal ) ) ).
% bot.extremum_uniqueI
thf(fact_405_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_406_b,axiom,
member_set_a @ ( sup_sup_set_a @ i @ ( set_ord_atLeast_a @ x ) ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ).
% b
thf(fact_407_Set_Ois__empty__def,axiom,
( is_empty_set_a
= ( ^ [A3: set_set_a] : ( A3 = bot_bot_set_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_408_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_409_boolean__algebra__cancel_Osup2,axiom,
! [B: sigma_measure_a,K: sigma_measure_a,B2: sigma_measure_a,A2: sigma_measure_a] :
( ( B
= ( sup_su27664952386392231sure_a @ K @ B2 ) )
=> ( ( sup_su27664952386392231sure_a @ A2 @ B )
= ( sup_su27664952386392231sure_a @ K @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_410_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_411_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_a,K: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( B
= ( sup_sup_set_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_set_a @ A2 @ B )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_412_boolean__algebra__cancel_Osup1,axiom,
! [A: sigma_measure_a,K: sigma_measure_a,A2: sigma_measure_a,B2: sigma_measure_a] :
( ( A
= ( sup_su27664952386392231sure_a @ K @ A2 ) )
=> ( ( sup_su27664952386392231sure_a @ A @ B2 )
= ( sup_su27664952386392231sure_a @ K @ ( sup_su27664952386392231sure_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_413_boolean__algebra__cancel_Osup1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_414_boolean__algebra__cancel_Osup1,axiom,
! [A: set_set_a,K: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( A
= ( sup_sup_set_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_set_a @ A @ B2 )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_415_subset__emptyI,axiom,
! [A: set_set_set_a] :
( ! [X2: set_set_a] :
~ ( member_set_set_a @ X2 @ A )
=> ( ord_le5722252365846178494_set_a @ A @ bot_bo3380559777022489994_set_a ) ) ).
% subset_emptyI
thf(fact_416_subset__emptyI,axiom,
! [A: set_set_a] :
( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_417_subset__emptyI,axiom,
! [A: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_418_subsetI,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( member_set_set_a @ X2 @ B ) )
=> ( ord_le5722252365846178494_set_a @ A @ B ) ) ).
% subsetI
thf(fact_419_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_420_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_421_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_422_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_423_in__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a,X3: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( member_set_set_a @ X3 @ A )
=> ( member_set_set_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_424_in__mono,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_425_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_426_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_427_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_428_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_429_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_430_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_431_subset__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
! [X: set_set_a] :
( ( member_set_set_a @ X @ A3 )
=> ( member_set_set_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_432_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_433_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_434_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_435_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_436_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_437_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_438_subset__iff,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
! [T: set_set_a] :
( ( member_set_set_a @ T @ A3 )
=> ( member_set_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_439_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_440_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_441_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_442_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_443_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_444_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_445_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_446_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_447_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z4: set_a] : ( Y4 = Z4 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_448_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z4: set_set_a] : ( Y4 = Z4 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_449_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_450_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_451_up__ray__borel,axiom,
! [I3: set_a] :
( ( up_ray_a @ I3 )
=> ( member_set_a @ I3 @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ).
% up_ray_borel
thf(fact_452_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_453_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_454_down__ray__borel,axiom,
! [I3: set_a] :
( ( down_ray_a @ I3 )
=> ( member_set_a @ I3 @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ).
% down_ray_borel
thf(fact_455_order__antisym__conv,axiom,
! [Y2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_456_order__antisym__conv,axiom,
! [Y2: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X3 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_457_order__antisym__conv,axiom,
! [Y2: sigma_measure_a,X3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ Y2 @ X3 )
=> ( ( ord_le254669795585780187sure_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_458_order__antisym__conv,axiom,
! [Y2: set_a > extend8495563244428889912nnreal,X3: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ Y2 @ X3 )
=> ( ( ord_le6700572704167691815nnreal @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_459_order__antisym__conv,axiom,
! [Y2: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y2 @ X3 )
=> ( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_460_order__antisym__conv,axiom,
! [Y2: a,X3: a] :
( ( ord_less_eq_a @ Y2 @ X3 )
=> ( ( ord_less_eq_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_461_linorder__le__cases,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ~ ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_462_linorder__le__cases,axiom,
! [X3: a,Y2: a] :
( ~ ( ord_less_eq_a @ X3 @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_463_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_464_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_465_ord__le__eq__subst,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > a,C: a] :
( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_466_ord__le__eq__subst,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_467_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_468_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > sigma_measure_a,C: sigma_measure_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_le254669795585780187sure_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le254669795585780187sure_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_469_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_470_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_471_ord__le__eq__subst,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a,F: sigma_measure_a > a,C: a] :
( ( ord_le254669795585780187sure_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: sigma_measure_a,Y3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_472_ord__le__eq__subst,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a,F: sigma_measure_a > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le254669795585780187sure_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: sigma_measure_a,Y3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_473_ord__eq__le__subst,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_474_ord__eq__le__subst,axiom,
! [A2: extend8495563244428889912nnreal,F: a > extend8495563244428889912nnreal,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_475_ord__eq__le__subst,axiom,
! [A2: a,F: extend8495563244428889912nnreal > a,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_476_ord__eq__le__subst,axiom,
! [A2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_477_ord__eq__le__subst,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_478_ord__eq__le__subst,axiom,
! [A2: sigma_measure_a,F: a > sigma_measure_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_le254669795585780187sure_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le254669795585780187sure_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_479_ord__eq__le__subst,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_480_ord__eq__le__subst,axiom,
! [A2: extend8495563244428889912nnreal,F: set_a > extend8495563244428889912nnreal,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_481_ord__eq__le__subst,axiom,
! [A2: a,F: sigma_measure_a > a,B2: sigma_measure_a,C: sigma_measure_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le254669795585780187sure_a @ B2 @ C )
=> ( ! [X2: sigma_measure_a,Y3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_482_ord__eq__le__subst,axiom,
! [A2: extend8495563244428889912nnreal,F: sigma_measure_a > extend8495563244428889912nnreal,B2: sigma_measure_a,C: sigma_measure_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le254669795585780187sure_a @ B2 @ C )
=> ( ! [X2: sigma_measure_a,Y3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_483_linorder__linear,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
| ( ord_le3935885782089961368nnreal @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_484_linorder__linear,axiom,
! [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
| ( ord_less_eq_a @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_485_order__eq__refl,axiom,
! [X3: set_a,Y2: set_a] :
( ( X3 = Y2 )
=> ( ord_less_eq_set_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_486_order__eq__refl,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( X3 = Y2 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_487_order__eq__refl,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a] :
( ( X3 = Y2 )
=> ( ord_le254669795585780187sure_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_488_order__eq__refl,axiom,
! [X3: set_a > extend8495563244428889912nnreal,Y2: set_a > extend8495563244428889912nnreal] :
( ( X3 = Y2 )
=> ( ord_le6700572704167691815nnreal @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_489_order__eq__refl,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( X3 = Y2 )
=> ( ord_le3935885782089961368nnreal @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_490_order__eq__refl,axiom,
! [X3: a,Y2: a] :
( ( X3 = Y2 )
=> ( ord_less_eq_a @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_491_order__subst2,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_492_order__subst2,axiom,
! [A2: a,B2: a,F: a > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_493_order__subst2,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > a,C: a] :
( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_494_order__subst2,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B2 ) @ C )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_495_order__subst2,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_496_order__subst2,axiom,
! [A2: a,B2: a,F: a > sigma_measure_a,C: sigma_measure_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_le254669795585780187sure_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_le254669795585780187sure_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le254669795585780187sure_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_497_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_498_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_499_order__subst2,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a,F: sigma_measure_a > a,C: a] :
( ( ord_le254669795585780187sure_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: sigma_measure_a,Y3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_500_order__subst2,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a,F: sigma_measure_a > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le254669795585780187sure_a @ A2 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B2 ) @ C )
=> ( ! [X2: sigma_measure_a,Y3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_501_order__subst1,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_502_order__subst1,axiom,
! [A2: a,F: extend8495563244428889912nnreal > a,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_503_order__subst1,axiom,
! [A2: extend8495563244428889912nnreal,F: a > extend8495563244428889912nnreal,B2: a,C: a] :
( ( ord_le3935885782089961368nnreal @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_504_order__subst1,axiom,
! [A2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_505_order__subst1,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_506_order__subst1,axiom,
! [A2: a,F: sigma_measure_a > a,B2: sigma_measure_a,C: sigma_measure_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le254669795585780187sure_a @ B2 @ C )
=> ( ! [X2: sigma_measure_a,Y3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_507_order__subst1,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_508_order__subst1,axiom,
! [A2: set_a,F: extend8495563244428889912nnreal > set_a,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_509_order__subst1,axiom,
! [A2: sigma_measure_a,F: a > sigma_measure_a,B2: a,C: a] :
( ( ord_le254669795585780187sure_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_le254669795585780187sure_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le254669795585780187sure_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_510_order__subst1,axiom,
! [A2: sigma_measure_a,F: extend8495563244428889912nnreal > sigma_measure_a,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le254669795585780187sure_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C )
=> ( ! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_le254669795585780187sure_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le254669795585780187sure_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_511_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z4: set_a] : ( Y4 = Z4 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_512_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z4: set_set_a] : ( Y4 = Z4 ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_513_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: sigma_measure_a,Z4: sigma_measure_a] : ( Y4 = Z4 ) )
= ( ^ [A5: sigma_measure_a,B5: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A5 @ B5 )
& ( ord_le254669795585780187sure_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_514_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a > extend8495563244428889912nnreal,Z4: set_a > extend8495563244428889912nnreal] : ( Y4 = Z4 ) )
= ( ^ [A5: set_a > extend8495563244428889912nnreal,B5: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A5 @ B5 )
& ( ord_le6700572704167691815nnreal @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_515_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z4: extend8495563244428889912nnreal] : ( Y4 = Z4 ) )
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A5 @ B5 )
& ( ord_le3935885782089961368nnreal @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_516_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z4: a] : ( Y4 = Z4 ) )
= ( ^ [A5: a,B5: a] :
( ( ord_less_eq_a @ A5 @ B5 )
& ( ord_less_eq_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_517_le__fun__def,axiom,
( ord_le6700572704167691815nnreal
= ( ^ [F2: set_a > extend8495563244428889912nnreal,G: set_a > extend8495563244428889912nnreal] :
! [X: set_a] : ( ord_le3935885782089961368nnreal @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_fun_def
thf(fact_518_le__funI,axiom,
! [F: set_a > extend8495563244428889912nnreal,G2: set_a > extend8495563244428889912nnreal] :
( ! [X2: set_a] : ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_le6700572704167691815nnreal @ F @ G2 ) ) ).
% le_funI
thf(fact_519_le__funE,axiom,
! [F: set_a > extend8495563244428889912nnreal,G2: set_a > extend8495563244428889912nnreal,X3: set_a] :
( ( ord_le6700572704167691815nnreal @ F @ G2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funE
thf(fact_520_le__funD,axiom,
! [F: set_a > extend8495563244428889912nnreal,G2: set_a > extend8495563244428889912nnreal,X3: set_a] :
( ( ord_le6700572704167691815nnreal @ F @ G2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funD
thf(fact_521_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_522_antisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_523_antisym,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ B2 )
=> ( ( ord_le254669795585780187sure_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_524_antisym,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A2 @ B2 )
=> ( ( ord_le6700572704167691815nnreal @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_525_antisym,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_526_antisym,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_527_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_528_dual__order_Otrans,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_529_dual__order_Otrans,axiom,
! [B2: sigma_measure_a,A2: sigma_measure_a,C: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ B2 @ A2 )
=> ( ( ord_le254669795585780187sure_a @ C @ B2 )
=> ( ord_le254669795585780187sure_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_530_dual__order_Otrans,axiom,
! [B2: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal,C: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ B2 @ A2 )
=> ( ( ord_le6700572704167691815nnreal @ C @ B2 )
=> ( ord_le6700572704167691815nnreal @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_531_dual__order_Otrans,axiom,
! [B2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ C @ B2 )
=> ( ord_le3935885782089961368nnreal @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_532_dual__order_Otrans,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ C @ B2 )
=> ( ord_less_eq_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_533_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_534_dual__order_Oantisym,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_535_dual__order_Oantisym,axiom,
! [B2: sigma_measure_a,A2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ B2 @ A2 )
=> ( ( ord_le254669795585780187sure_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_536_dual__order_Oantisym,axiom,
! [B2: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ B2 @ A2 )
=> ( ( ord_le6700572704167691815nnreal @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_537_dual__order_Oantisym,axiom,
! [B2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_538_dual__order_Oantisym,axiom,
! [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_539_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z4: set_a] : ( Y4 = Z4 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_540_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_a,Z4: set_set_a] : ( Y4 = Z4 ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
& ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_541_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: sigma_measure_a,Z4: sigma_measure_a] : ( Y4 = Z4 ) )
= ( ^ [A5: sigma_measure_a,B5: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ B5 @ A5 )
& ( ord_le254669795585780187sure_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_542_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a > extend8495563244428889912nnreal,Z4: set_a > extend8495563244428889912nnreal] : ( Y4 = Z4 ) )
= ( ^ [A5: set_a > extend8495563244428889912nnreal,B5: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ B5 @ A5 )
& ( ord_le6700572704167691815nnreal @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_543_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z4: extend8495563244428889912nnreal] : ( Y4 = Z4 ) )
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B5 @ A5 )
& ( ord_le3935885782089961368nnreal @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_544_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: a,Z4: a] : ( Y4 = Z4 ) )
= ( ^ [A5: a,B5: a] :
( ( ord_less_eq_a @ B5 @ A5 )
& ( ord_less_eq_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_545_linorder__wlog,axiom,
! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ! [A6: extend8495563244428889912nnreal,B6: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: extend8495563244428889912nnreal,B6: extend8495563244428889912nnreal] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_546_linorder__wlog,axiom,
! [P: a > a > $o,A2: a,B2: a] :
( ! [A6: a,B6: a] :
( ( ord_less_eq_a @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: a,B6: a] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_547_order__trans,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z )
=> ( ord_less_eq_set_a @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_548_order__trans,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ Z )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_549_order__trans,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a,Z: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X3 @ Y2 )
=> ( ( ord_le254669795585780187sure_a @ Y2 @ Z )
=> ( ord_le254669795585780187sure_a @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_550_order__trans,axiom,
! [X3: set_a > extend8495563244428889912nnreal,Y2: set_a > extend8495563244428889912nnreal,Z: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ X3 @ Y2 )
=> ( ( ord_le6700572704167691815nnreal @ Y2 @ Z )
=> ( ord_le6700572704167691815nnreal @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_551_order__trans,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ( ord_le3935885782089961368nnreal @ Y2 @ Z )
=> ( ord_le3935885782089961368nnreal @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_552_order__trans,axiom,
! [X3: a,Y2: a,Z: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z )
=> ( ord_less_eq_a @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_553_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_554_order_Otrans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_555_order_Otrans,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a,C: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ B2 )
=> ( ( ord_le254669795585780187sure_a @ B2 @ C )
=> ( ord_le254669795585780187sure_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_556_order_Otrans,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal,C: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A2 @ B2 )
=> ( ( ord_le6700572704167691815nnreal @ B2 @ C )
=> ( ord_le6700572704167691815nnreal @ A2 @ C ) ) ) ).
% order.trans
thf(fact_557_order_Otrans,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C )
=> ( ord_le3935885782089961368nnreal @ A2 @ C ) ) ) ).
% order.trans
thf(fact_558_order_Otrans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_559_order__antisym,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_560_order__antisym,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_561_order__antisym,axiom,
! [X3: sigma_measure_a,Y2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X3 @ Y2 )
=> ( ( ord_le254669795585780187sure_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_562_order__antisym,axiom,
! [X3: set_a > extend8495563244428889912nnreal,Y2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ X3 @ Y2 )
=> ( ( ord_le6700572704167691815nnreal @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_563_order__antisym,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ( ord_le3935885782089961368nnreal @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_564_order__antisym,axiom,
! [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_565_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_566_ord__le__eq__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_567_ord__le__eq__trans,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a,C: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le254669795585780187sure_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_568_ord__le__eq__trans,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal,C: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le6700572704167691815nnreal @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_569_ord__le__eq__trans,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le3935885782089961368nnreal @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_570_ord__le__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_571_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_572_ord__eq__le__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A2 = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_573_ord__eq__le__trans,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a,C: sigma_measure_a] :
( ( A2 = B2 )
=> ( ( ord_le254669795585780187sure_a @ B2 @ C )
=> ( ord_le254669795585780187sure_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_574_ord__eq__le__trans,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal,C: set_a > extend8495563244428889912nnreal] :
( ( A2 = B2 )
=> ( ( ord_le6700572704167691815nnreal @ B2 @ C )
=> ( ord_le6700572704167691815nnreal @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_575_ord__eq__le__trans,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A2 = B2 )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C )
=> ( ord_le3935885782089961368nnreal @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_576_ord__eq__le__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_577_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z4: set_a] : ( Y4 = Z4 ) )
= ( ^ [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_578_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z4: set_set_a] : ( Y4 = Z4 ) )
= ( ^ [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
& ( ord_le3724670747650509150_set_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_579_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: sigma_measure_a,Z4: sigma_measure_a] : ( Y4 = Z4 ) )
= ( ^ [X: sigma_measure_a,Y: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ X @ Y )
& ( ord_le254669795585780187sure_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_580_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a > extend8495563244428889912nnreal,Z4: set_a > extend8495563244428889912nnreal] : ( Y4 = Z4 ) )
= ( ^ [X: set_a > extend8495563244428889912nnreal,Y: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ X @ Y )
& ( ord_le6700572704167691815nnreal @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_581_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z4: extend8495563244428889912nnreal] : ( Y4 = Z4 ) )
= ( ^ [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y )
& ( ord_le3935885782089961368nnreal @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_582_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z4: a] : ( Y4 = Z4 ) )
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( ord_less_eq_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_583_le__cases3,axiom,
! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ~ ( ord_le3935885782089961368nnreal @ Y2 @ Z ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y2 @ X3 )
=> ~ ( ord_le3935885782089961368nnreal @ X3 @ Z ) )
=> ( ( ( ord_le3935885782089961368nnreal @ X3 @ Z )
=> ~ ( ord_le3935885782089961368nnreal @ Z @ Y2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Z @ Y2 )
=> ~ ( ord_le3935885782089961368nnreal @ Y2 @ X3 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y2 @ Z )
=> ~ ( ord_le3935885782089961368nnreal @ Z @ X3 ) )
=> ~ ( ( ord_le3935885782089961368nnreal @ Z @ X3 )
=> ~ ( ord_le3935885782089961368nnreal @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_584_le__cases3,axiom,
! [X3: a,Y2: a,Z: a] :
( ( ( ord_less_eq_a @ X3 @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_a @ Y2 @ X3 )
=> ~ ( ord_less_eq_a @ X3 @ Z ) )
=> ( ( ( ord_less_eq_a @ X3 @ Z )
=> ~ ( ord_less_eq_a @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_a @ Z @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_a @ Y2 @ Z )
=> ~ ( ord_less_eq_a @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_a @ Z @ X3 )
=> ~ ( ord_less_eq_a @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_585_nle__le,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B2 ) )
= ( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_586_nle__le,axiom,
! [A2: a,B2: a] :
( ( ~ ( ord_less_eq_a @ A2 @ B2 ) )
= ( ( ord_less_eq_a @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_587_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_588_bot__fun__def,axiom,
( bot_bot_set_a_o
= ( ^ [X: set_a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_589_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_590_sets_OUn,axiom,
! [A2: set_set_a,M: sigma_measure_set_a,B2: set_set_a] :
( ( member_set_set_a @ A2 @ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a @ B2 @ ( sigma_sets_set_a @ M ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( sigma_sets_set_a @ M ) ) ) ) ).
% sets.Un
thf(fact_591_sets_Oempty__sets,axiom,
! [M: sigma_measure_set_a] : ( member_set_set_a @ bot_bot_set_set_a @ ( sigma_sets_set_a @ M ) ) ).
% sets.empty_sets
thf(fact_592_sets_Oempty__sets,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_sets_a @ M ) ) ).
% sets.empty_sets
thf(fact_593_atLeast__borel,axiom,
! [A2: a] : ( member_set_a @ ( set_ord_atLeast_a @ A2 ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% atLeast_borel
thf(fact_594_atMost__borel,axiom,
! [A2: a] : ( member_set_a @ ( set_ord_atMost_a @ A2 ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% atMost_borel
thf(fact_595_Greatest__equality,axiom,
! [P: set_a > $o,X3: set_a] :
( ( P @ X3 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X3 ) )
=> ( ( order_Greatest_set_a @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_596_Greatest__equality,axiom,
! [P: set_set_a > $o,X3: set_set_a] :
( ( P @ X3 )
=> ( ! [Y3: set_set_a] :
( ( P @ Y3 )
=> ( ord_le3724670747650509150_set_a @ Y3 @ X3 ) )
=> ( ( order_3565860530148683671_set_a @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_597_Greatest__equality,axiom,
! [P: sigma_measure_a > $o,X3: sigma_measure_a] :
( ( P @ X3 )
=> ( ! [Y3: sigma_measure_a] :
( ( P @ Y3 )
=> ( ord_le254669795585780187sure_a @ Y3 @ X3 ) )
=> ( ( order_4873613539232846306sure_a @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_598_Greatest__equality,axiom,
! [P: ( set_a > extend8495563244428889912nnreal ) > $o,X3: set_a > extend8495563244428889912nnreal] :
( ( P @ X3 )
=> ( ! [Y3: set_a > extend8495563244428889912nnreal] :
( ( P @ Y3 )
=> ( ord_le6700572704167691815nnreal @ Y3 @ X3 ) )
=> ( ( order_1563119970664660704nnreal @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_599_Greatest__equality,axiom,
! [P: extend8495563244428889912nnreal > $o,X3: extend8495563244428889912nnreal] :
( ( P @ X3 )
=> ( ! [Y3: extend8495563244428889912nnreal] :
( ( P @ Y3 )
=> ( ord_le3935885782089961368nnreal @ Y3 @ X3 ) )
=> ( ( order_7545170809120406815nnreal @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_600_Greatest__equality,axiom,
! [P: a > $o,X3: a] :
( ( P @ X3 )
=> ( ! [Y3: a] :
( ( P @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X3 ) )
=> ( ( order_Greatest_a @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_601_GreatestI2__order,axiom,
! [P: set_a > $o,X3: set_a,Q: set_a > $o] :
( ( P @ X3 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X3 ) )
=> ( ! [X2: set_a] :
( ( P @ X2 )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_602_GreatestI2__order,axiom,
! [P: set_set_a > $o,X3: set_set_a,Q: set_set_a > $o] :
( ( P @ X3 )
=> ( ! [Y3: set_set_a] :
( ( P @ Y3 )
=> ( ord_le3724670747650509150_set_a @ Y3 @ X3 ) )
=> ( ! [X2: set_set_a] :
( ( P @ X2 )
=> ( ! [Y5: set_set_a] :
( ( P @ Y5 )
=> ( ord_le3724670747650509150_set_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_3565860530148683671_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_603_GreatestI2__order,axiom,
! [P: sigma_measure_a > $o,X3: sigma_measure_a,Q: sigma_measure_a > $o] :
( ( P @ X3 )
=> ( ! [Y3: sigma_measure_a] :
( ( P @ Y3 )
=> ( ord_le254669795585780187sure_a @ Y3 @ X3 ) )
=> ( ! [X2: sigma_measure_a] :
( ( P @ X2 )
=> ( ! [Y5: sigma_measure_a] :
( ( P @ Y5 )
=> ( ord_le254669795585780187sure_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_4873613539232846306sure_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_604_GreatestI2__order,axiom,
! [P: ( set_a > extend8495563244428889912nnreal ) > $o,X3: set_a > extend8495563244428889912nnreal,Q: ( set_a > extend8495563244428889912nnreal ) > $o] :
( ( P @ X3 )
=> ( ! [Y3: set_a > extend8495563244428889912nnreal] :
( ( P @ Y3 )
=> ( ord_le6700572704167691815nnreal @ Y3 @ X3 ) )
=> ( ! [X2: set_a > extend8495563244428889912nnreal] :
( ( P @ X2 )
=> ( ! [Y5: set_a > extend8495563244428889912nnreal] :
( ( P @ Y5 )
=> ( ord_le6700572704167691815nnreal @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1563119970664660704nnreal @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_605_GreatestI2__order,axiom,
! [P: extend8495563244428889912nnreal > $o,X3: extend8495563244428889912nnreal,Q: extend8495563244428889912nnreal > $o] :
( ( P @ X3 )
=> ( ! [Y3: extend8495563244428889912nnreal] :
( ( P @ Y3 )
=> ( ord_le3935885782089961368nnreal @ Y3 @ X3 ) )
=> ( ! [X2: extend8495563244428889912nnreal] :
( ( P @ X2 )
=> ( ! [Y5: extend8495563244428889912nnreal] :
( ( P @ Y5 )
=> ( ord_le3935885782089961368nnreal @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_7545170809120406815nnreal @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_606_GreatestI2__order,axiom,
! [P: a > $o,X3: a,Q: a > $o] :
( ( P @ X3 )
=> ( ! [Y3: a] :
( ( P @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X3 ) )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_607_Icc__subset__Ici__iff,axiom,
! [L: set_set_a,H: set_set_a,L2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ L @ H ) @ ( set_or3904034815786525833_set_a @ L2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ L @ H )
| ( ord_le3724670747650509150_set_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_608_Icc__subset__Ici__iff,axiom,
! [L: sigma_measure_a,H: sigma_measure_a,L2: sigma_measure_a] :
( ( ord_le2601950661898901649sure_a @ ( set_or8965261581713928454sure_a @ L @ H ) @ ( set_or6558828460305013104sure_a @ L2 ) )
= ( ~ ( ord_le254669795585780187sure_a @ L @ H )
| ( ord_le254669795585780187sure_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_609_Icc__subset__Ici__iff,axiom,
! [L: set_a > extend8495563244428889912nnreal,H: set_a > extend8495563244428889912nnreal,L2: set_a > extend8495563244428889912nnreal] :
( ( ord_le2718372912061172615nnreal @ ( set_or5554580835743624380nnreal @ L @ H ) @ ( set_or3682978270108489938nnreal @ L2 ) )
= ( ~ ( ord_le6700572704167691815nnreal @ L @ H )
| ( ord_le6700572704167691815nnreal @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_610_Icc__subset__Ici__iff,axiom,
! [L: extend8495563244428889912nnreal,H: extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ ( set_or6180253510304638531nnreal @ L @ H ) @ ( set_or8062060312790847405nnreal @ L2 ) )
= ( ~ ( ord_le3935885782089961368nnreal @ L @ H )
| ( ord_le3935885782089961368nnreal @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_611_Icc__subset__Ici__iff,axiom,
! [L: set_a,H: set_a,L2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ L @ H ) @ ( set_or8362275514725411625_set_a @ L2 ) )
= ( ~ ( ord_less_eq_set_a @ L @ H )
| ( ord_less_eq_set_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_612_Icc__subset__Ici__iff,axiom,
! [L: a,H: a,L2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ L @ H ) @ ( set_ord_atLeast_a @ L2 ) )
= ( ~ ( ord_less_eq_a @ L @ H )
| ( ord_less_eq_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_613_Icc__eq__Icc,axiom,
! [L: set_set_a,H: set_set_a,L2: set_set_a,H2: set_set_a] :
( ( ( set_or4761464488706262899_set_a @ L @ H )
= ( set_or4761464488706262899_set_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le3724670747650509150_set_a @ L @ H )
& ~ ( ord_le3724670747650509150_set_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_614_Icc__eq__Icc,axiom,
! [L: sigma_measure_a,H: sigma_measure_a,L2: sigma_measure_a,H2: sigma_measure_a] :
( ( ( set_or8965261581713928454sure_a @ L @ H )
= ( set_or8965261581713928454sure_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le254669795585780187sure_a @ L @ H )
& ~ ( ord_le254669795585780187sure_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_615_Icc__eq__Icc,axiom,
! [L: set_a > extend8495563244428889912nnreal,H: set_a > extend8495563244428889912nnreal,L2: set_a > extend8495563244428889912nnreal,H2: set_a > extend8495563244428889912nnreal] :
( ( ( set_or5554580835743624380nnreal @ L @ H )
= ( set_or5554580835743624380nnreal @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le6700572704167691815nnreal @ L @ H )
& ~ ( ord_le6700572704167691815nnreal @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_616_Icc__eq__Icc,axiom,
! [L: extend8495563244428889912nnreal,H: extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal,H2: extend8495563244428889912nnreal] :
( ( ( set_or6180253510304638531nnreal @ L @ H )
= ( set_or6180253510304638531nnreal @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le3935885782089961368nnreal @ L @ H )
& ~ ( ord_le3935885782089961368nnreal @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_617_Icc__eq__Icc,axiom,
! [L: set_a,H: set_a,L2: set_a,H2: set_a] :
( ( ( set_or6288561110385358355_set_a @ L @ H )
= ( set_or6288561110385358355_set_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_a @ L @ H )
& ~ ( ord_less_eq_set_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_618_Icc__eq__Icc,axiom,
! [L: a,H: a,L2: a,H2: a] :
( ( ( set_or672772299803893939Most_a @ L @ H )
= ( set_or672772299803893939Most_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_a @ L @ H )
& ~ ( ord_less_eq_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_619_atLeastAtMost__iff,axiom,
! [I2: set_set_a,L: set_set_a,U: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or4761464488706262899_set_a @ L @ U ) )
= ( ( ord_le3724670747650509150_set_a @ L @ I2 )
& ( ord_le3724670747650509150_set_a @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_620_atLeastAtMost__iff,axiom,
! [I2: sigma_measure_a,L: sigma_measure_a,U: sigma_measure_a] :
( ( member3534519376729797778sure_a @ I2 @ ( set_or8965261581713928454sure_a @ L @ U ) )
= ( ( ord_le254669795585780187sure_a @ L @ I2 )
& ( ord_le254669795585780187sure_a @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_621_atLeastAtMost__iff,axiom,
! [I2: set_a > extend8495563244428889912nnreal,L: set_a > extend8495563244428889912nnreal,U: set_a > extend8495563244428889912nnreal] :
( ( member4180043592386426928nnreal @ I2 @ ( set_or5554580835743624380nnreal @ L @ U ) )
= ( ( ord_le6700572704167691815nnreal @ L @ I2 )
& ( ord_le6700572704167691815nnreal @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_622_atLeastAtMost__iff,axiom,
! [I2: extend8495563244428889912nnreal,L: extend8495563244428889912nnreal,U: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I2 @ ( set_or6180253510304638531nnreal @ L @ U ) )
= ( ( ord_le3935885782089961368nnreal @ L @ I2 )
& ( ord_le3935885782089961368nnreal @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_623_atLeastAtMost__iff,axiom,
! [I2: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I2 @ ( set_or6288561110385358355_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_624_atLeastAtMost__iff,axiom,
! [I2: a,L: a,U: a] :
( ( member_a @ I2 @ ( set_or672772299803893939Most_a @ L @ U ) )
= ( ( ord_less_eq_a @ L @ I2 )
& ( ord_less_eq_a @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_625_atLeastatMost__subset__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a,D2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ A2 @ B2 ) @ ( set_or4761464488706262899_set_a @ C @ D2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 )
| ( ( ord_le3724670747650509150_set_a @ C @ A2 )
& ( ord_le3724670747650509150_set_a @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_626_atLeastatMost__subset__iff,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a,C: sigma_measure_a,D2: sigma_measure_a] :
( ( ord_le2601950661898901649sure_a @ ( set_or8965261581713928454sure_a @ A2 @ B2 ) @ ( set_or8965261581713928454sure_a @ C @ D2 ) )
= ( ~ ( ord_le254669795585780187sure_a @ A2 @ B2 )
| ( ( ord_le254669795585780187sure_a @ C @ A2 )
& ( ord_le254669795585780187sure_a @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_627_atLeastatMost__subset__iff,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal,C: set_a > extend8495563244428889912nnreal,D2: set_a > extend8495563244428889912nnreal] :
( ( ord_le2718372912061172615nnreal @ ( set_or5554580835743624380nnreal @ A2 @ B2 ) @ ( set_or5554580835743624380nnreal @ C @ D2 ) )
= ( ~ ( ord_le6700572704167691815nnreal @ A2 @ B2 )
| ( ( ord_le6700572704167691815nnreal @ C @ A2 )
& ( ord_le6700572704167691815nnreal @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_628_atLeastatMost__subset__iff,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ ( set_or6180253510304638531nnreal @ A2 @ B2 ) @ ( set_or6180253510304638531nnreal @ C @ D2 ) )
= ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B2 )
| ( ( ord_le3935885782089961368nnreal @ C @ A2 )
& ( ord_le3935885782089961368nnreal @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_629_atLeastatMost__subset__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or6288561110385358355_set_a @ C @ D2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_630_atLeastatMost__subset__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or672772299803893939Most_a @ C @ D2 ) )
= ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_631_atLeastatMost__empty__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( set_or4761464488706262899_set_a @ A2 @ B2 )
= bot_bo3380559777022489994_set_a )
= ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_632_atLeastatMost__empty__iff,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a] :
( ( ( set_or8965261581713928454sure_a @ A2 @ B2 )
= bot_bo5171090233048072389sure_a )
= ( ~ ( ord_le254669795585780187sure_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_633_atLeastatMost__empty__iff,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal] :
( ( ( set_or5554580835743624380nnreal @ A2 @ B2 )
= bot_bo8079028080804873299nnreal )
= ( ~ ( ord_le6700572704167691815nnreal @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_634_atLeastatMost__empty__iff,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ( set_or6180253510304638531nnreal @ A2 @ B2 )
= bot_bo4854962954004695426nnreal )
= ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_635_atLeastatMost__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( set_or6288561110385358355_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_636_atLeastatMost__empty__iff,axiom,
! [A2: a,B2: a] :
( ( ( set_or672772299803893939Most_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ~ ( ord_less_eq_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_637_atLeastatMost__empty__iff2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( bot_bo3380559777022489994_set_a
= ( set_or4761464488706262899_set_a @ A2 @ B2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_638_atLeastatMost__empty__iff2,axiom,
! [A2: sigma_measure_a,B2: sigma_measure_a] :
( ( bot_bo5171090233048072389sure_a
= ( set_or8965261581713928454sure_a @ A2 @ B2 ) )
= ( ~ ( ord_le254669795585780187sure_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_639_atLeastatMost__empty__iff2,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B2: set_a > extend8495563244428889912nnreal] :
( ( bot_bo8079028080804873299nnreal
= ( set_or5554580835743624380nnreal @ A2 @ B2 ) )
= ( ~ ( ord_le6700572704167691815nnreal @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_640_atLeastatMost__empty__iff2,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( bot_bo4854962954004695426nnreal
= ( set_or6180253510304638531nnreal @ A2 @ B2 ) )
= ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_641_atLeastatMost__empty__iff2,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_set_a
= ( set_or6288561110385358355_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_642_atLeastatMost__empty__iff2,axiom,
! [A2: a,B2: a] :
( ( bot_bot_set_a
= ( set_or672772299803893939Most_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_643_Icc__subset__Iic__iff,axiom,
! [L: set_a > extend8495563244428889912nnreal,H: set_a > extend8495563244428889912nnreal,H2: set_a > extend8495563244428889912nnreal] :
( ( ord_le2718372912061172615nnreal @ ( set_or5554580835743624380nnreal @ L @ H ) @ ( set_or7674582181447345870nnreal @ H2 ) )
= ( ~ ( ord_le6700572704167691815nnreal @ L @ H )
| ( ord_le6700572704167691815nnreal @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_644_Icc__subset__Iic__iff,axiom,
! [L: extend8495563244428889912nnreal,H: extend8495563244428889912nnreal,H2: extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ ( set_or6180253510304638531nnreal @ L @ H ) @ ( set_or4252868341878644913nnreal @ H2 ) )
= ( ~ ( ord_le3935885782089961368nnreal @ L @ H )
| ( ord_le3935885782089961368nnreal @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_645_Icc__subset__Iic__iff,axiom,
! [L: set_a,H: set_a,H2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ L @ H ) @ ( set_ord_atMost_set_a @ H2 ) )
= ( ~ ( ord_less_eq_set_a @ L @ H )
| ( ord_less_eq_set_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_646_Icc__subset__Iic__iff,axiom,
! [L: a,H: a,H2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ L @ H ) @ ( set_ord_atMost_a @ H2 ) )
= ( ~ ( ord_less_eq_a @ L @ H )
| ( ord_less_eq_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_647_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_648_ivl__disj__un__two__touch_I4_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_649_atLeastatMost__empty_H,axiom,
! [A2: a,B2: a] :
( ~ ( ord_less_eq_a @ A2 @ B2 )
=> ( ( set_or672772299803893939Most_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% atLeastatMost_empty'
thf(fact_650_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X: set_a] : ( member_set_a @ X @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_651_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_652_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_653_ivl__disj__un__one_I7_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ U ) @ ( set_or8632414552788122084Than_a @ U ) )
= ( set_ord_atLeast_a @ L ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_654_ivl__disj__un__one_I4_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or672772299803893939Most_a @ L @ U ) )
= ( set_ord_atMost_a @ U ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_655_atLeastAtMost__def,axiom,
( set_or672772299803893939Most_a
= ( ^ [L3: a,U2: a] : ( inf_inf_set_a @ ( set_ord_atLeast_a @ L3 ) @ ( set_ord_atMost_a @ U2 ) ) ) ) ).
% atLeastAtMost_def
thf(fact_656_sets__Ball,axiom,
! [I3: set_a,A: a > set_a,M: a > sigma_measure_a,I2: a] :
( ! [X2: a] :
( ( member_a @ X2 @ I3 )
=> ( member_set_a @ ( A @ X2 ) @ ( sigma_sets_a @ ( M @ X2 ) ) ) )
=> ( ( member_a @ I2 @ I3 )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_657_sets__Ball,axiom,
! [I3: set_set_a,A: set_a > set_a,M: set_a > sigma_measure_a,I2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ I3 )
=> ( member_set_a @ ( A @ X2 ) @ ( sigma_sets_a @ ( M @ X2 ) ) ) )
=> ( ( member_set_a @ I2 @ I3 )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_658_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_659_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_660_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_661_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_662_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ X3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_663_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X3 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_664_inf__bot__right,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ X3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_665_inf__bot__left,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X3 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_666_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_667_inf__sup__absorb,axiom,
! [X3: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_668_inf__sup__absorb,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ X3 @ Y2 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_669_sup__inf__absorb,axiom,
! [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ X3 @ Y2 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_670_sup__inf__absorb,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ X3 @ Y2 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_671_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_672_Int__Un__eq_I4_J,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( sup_sup_set_set_a @ T2 @ ( inf_inf_set_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_673_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_674_Int__Un__eq_I3_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ S @ ( inf_inf_set_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_675_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_676_Int__Un__eq_I2_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_677_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_678_Int__Un__eq_I1_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_679_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_680_Un__Int__eq_I4_J,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( inf_inf_set_set_a @ T2 @ ( sup_sup_set_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_681_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_682_Un__Int__eq_I3_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ S @ ( sup_sup_set_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_683_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_684_Un__Int__eq_I2_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_685_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_686_Un__Int__eq_I1_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_687_lessThan__subset__iff,axiom,
! [X3: a,Y2: a] :
( ( ord_less_eq_set_a @ ( set_ord_lessThan_a @ X3 ) @ ( set_ord_lessThan_a @ Y2 ) )
= ( ord_less_eq_a @ X3 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_688_greaterThan__subset__iff,axiom,
! [X3: a,Y2: a] :
( ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ X3 ) @ ( set_or8632414552788122084Than_a @ Y2 ) )
= ( ord_less_eq_a @ Y2 @ X3 ) ) ).
% greaterThan_subset_iff
thf(fact_689_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_690_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_691_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_692_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_693_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_694_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_695_ivl__disj__int__one_I4_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or672772299803893939Most_a @ L @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(4)
thf(fact_696_ivl__disj__int__one_I7_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_or672772299803893939Most_a @ L @ U ) @ ( set_or8632414552788122084Than_a @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(7)
thf(fact_697_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y2 ) @ ( inf_inf_set_a @ X3 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_698_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y2 @ Z ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y2 ) @ ( inf_inf_set_set_a @ X3 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_699_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y2 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_700_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y2 @ Z ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y2 ) @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_701_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_a,Z: set_a,X3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y2 @ Z ) @ X3 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y2 @ X3 ) @ ( inf_inf_set_a @ Z @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_702_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_set_a,Z: set_set_a,X3: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y2 @ Z ) @ X3 )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y2 @ X3 ) @ ( inf_inf_set_set_a @ Z @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_703_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_a,Z: set_a,X3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y2 @ Z ) @ X3 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y2 @ X3 ) @ ( sup_sup_set_a @ Z @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_704_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_set_a,Z: set_set_a,X3: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y2 @ Z ) @ X3 )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y2 @ X3 ) @ ( sup_sup_set_set_a @ Z @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_705_distrib__imp1,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y2 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ ( sup_sup_set_a @ X3 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_706_distrib__imp1,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ! [X2: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ ( inf_inf_set_set_a @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y2 @ Z ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y2 ) @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_707_distrib__imp2,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y2 ) @ ( inf_inf_set_a @ X3 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_708_distrib__imp2,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ! [X2: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y3 ) @ ( sup_sup_set_set_a @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y2 @ Z ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y2 ) @ ( inf_inf_set_set_a @ X3 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_709_inf__sup__distrib1,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y2 ) @ ( inf_inf_set_a @ X3 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_710_inf__sup__distrib1,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y2 @ Z ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y2 ) @ ( inf_inf_set_set_a @ X3 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_711_inf__sup__distrib2,axiom,
! [Y2: set_a,Z: set_a,X3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y2 @ Z ) @ X3 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y2 @ X3 ) @ ( inf_inf_set_a @ Z @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_712_inf__sup__distrib2,axiom,
! [Y2: set_set_a,Z: set_set_a,X3: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y2 @ Z ) @ X3 )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y2 @ X3 ) @ ( inf_inf_set_set_a @ Z @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_713_sup__inf__distrib1,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y2 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_714_sup__inf__distrib1,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y2 @ Z ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y2 ) @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_715_sup__inf__distrib2,axiom,
! [Y2: set_a,Z: set_a,X3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y2 @ Z ) @ X3 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y2 @ X3 ) @ ( sup_sup_set_a @ Z @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_716_sup__inf__distrib2,axiom,
! [Y2: set_set_a,Z: set_set_a,X3: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y2 @ Z ) @ X3 )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y2 @ X3 ) @ ( sup_sup_set_set_a @ Z @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_717_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( 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_718_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 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( 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_719_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ! [Y: a] :
( ( member_a @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_720_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_721_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_722_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ~ ( member_set_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_723_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_724_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ~ ( member_set_a @ X2 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_725_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_726_Un__Int__crazy,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ B @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ B @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_727_Un__Int__crazy,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ B @ C2 ) ) @ ( inf_inf_set_set_a @ C2 @ A ) )
= ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ B @ C2 ) ) @ ( sup_sup_set_set_a @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_728_Int__Un__distrib,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_729_Int__Un__distrib,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_730_Un__Int__distrib,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_731_Un__Int__distrib,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_732_Int__Un__distrib2,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A ) @ ( inf_inf_set_a @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_733_Int__Un__distrib2,axiom,
! [B: set_set_a,C2: set_set_a,A: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ B @ C2 ) @ A )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ B @ A ) @ ( inf_inf_set_set_a @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_734_Un__Int__distrib2,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C2 ) @ A )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_735_Un__Int__distrib2,axiom,
! [B: set_set_a,C2: set_set_a,A: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ B @ C2 ) @ A )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ B @ A ) @ ( sup_sup_set_set_a @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_736_distrib__inf__le,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y2 ) @ ( inf_inf_set_a @ X3 @ Z ) ) @ ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_737_distrib__inf__le,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y2 ) @ ( inf_inf_set_set_a @ X3 @ Z ) ) @ ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_738_distrib__sup__le,axiom,
! [X3: set_a,Y2: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y2 @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_739_distrib__sup__le,axiom,
! [X3: set_set_a,Y2: set_set_a,Z: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y2 @ Z ) ) @ ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y2 ) @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_740_lessThan__borel,axiom,
! [A2: a] : ( member_set_a @ ( set_ord_lessThan_a @ A2 ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% lessThan_borel
thf(fact_741_greaterThan__borel,axiom,
! [A2: a] : ( member_set_a @ ( set_or8632414552788122084Than_a @ A2 ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% greaterThan_borel
thf(fact_742_Ioi__le__Ico,axiom,
! [A2: a] : ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ A2 ) @ ( set_ord_atLeast_a @ A2 ) ) ).
% Ioi_le_Ico
thf(fact_743_Un__Int__assoc__eq,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_744_Un__Int__assoc__eq,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) ) )
= ( ord_le3724670747650509150_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_745_closed__cdi__Un,axiom,
! [Omega: set_set_a,M: set_set_set_a,A: set_set_a,B: set_set_a] :
( ( sigma_8605711284820113968_set_a @ Omega @ M )
=> ( ( member_set_set_a @ bot_bot_set_set_a @ M )
=> ( ( member_set_set_a @ A @ M )
=> ( ( member_set_set_a @ B @ M )
=> ( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ M ) ) ) ) ) ) ).
% closed_cdi_Un
thf(fact_746_closed__cdi__Un,axiom,
! [Omega: set_a,M: set_set_a,A: set_a,B: set_a] :
( ( sigma_closed_cdi_a @ Omega @ M )
=> ( ( member_set_a @ bot_bot_set_a @ M )
=> ( ( member_set_a @ A @ M )
=> ( ( member_set_a @ B @ M )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( member_set_a @ ( sup_sup_set_a @ A @ B ) @ M ) ) ) ) ) ) ).
% closed_cdi_Un
thf(fact_747_sets_Ogenerated__ring__disjoint__Un,axiom,
! [A2: set_set_a,M: sigma_measure_set_a,B2: set_set_a] :
( ( member_set_set_a @ A2 @ ( sigma_7997604194779675523_set_a @ ( sigma_sets_set_a @ M ) ) )
=> ( ( member_set_set_a @ B2 @ ( sigma_7997604194779675523_set_a @ ( sigma_sets_set_a @ M ) ) )
=> ( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( sigma_7997604194779675523_set_a @ ( sigma_sets_set_a @ M ) ) ) ) ) ) ).
% sets.generated_ring_disjoint_Un
thf(fact_748_sets_Ogenerated__ring__disjoint__Un,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_6990475660951004707ring_a @ ( sigma_sets_a @ M ) ) )
=> ( ( member_set_a @ B2 @ ( sigma_6990475660951004707ring_a @ ( sigma_sets_a @ M ) ) )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sigma_6990475660951004707ring_a @ ( sigma_sets_a @ M ) ) ) ) ) ) ).
% sets.generated_ring_disjoint_Un
thf(fact_749_ivl__disj__un__singleton_I1_J,axiom,
! [L: a] :
( ( sup_sup_set_a @ ( insert_a @ L @ bot_bot_set_a ) @ ( set_or8632414552788122084Than_a @ L ) )
= ( set_ord_atLeast_a @ L ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_750_atMost__Int__atLeast,axiom,
! [N: a] :
( ( inf_inf_set_a @ ( set_ord_atMost_a @ N ) @ ( set_ord_atLeast_a @ N ) )
= ( insert_a @ N @ bot_bot_set_a ) ) ).
% atMost_Int_atLeast
thf(fact_751_ivl__disj__un__singleton_I2_J,axiom,
! [U: a] :
( ( sup_sup_set_a @ ( set_ord_lessThan_a @ U ) @ ( insert_a @ U @ bot_bot_set_a ) )
= ( set_ord_atMost_a @ U ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_752_Int__atLeastAtMostR2,axiom,
! [A2: a,C: a,D2: a] :
( ( inf_inf_set_a @ ( set_ord_atLeast_a @ A2 ) @ ( set_or672772299803893939Most_a @ C @ D2 ) )
= ( set_or672772299803893939Most_a @ ( ord_max_a @ A2 @ C ) @ D2 ) ) ).
% Int_atLeastAtMostR2
thf(fact_753_Int__atLeastAtMostL2,axiom,
! [A2: a,B2: a,C: a] :
( ( inf_inf_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_ord_atLeast_a @ C ) )
= ( set_or672772299803893939Most_a @ ( ord_max_a @ A2 @ C ) @ B2 ) ) ).
% Int_atLeastAtMostL2
thf(fact_754_Int__atLeastAtMostL1,axiom,
! [A2: a,B2: a,D2: a] :
( ( inf_inf_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_ord_atMost_a @ D2 ) )
= ( set_or672772299803893939Most_a @ A2 @ ( ord_min_a @ B2 @ D2 ) ) ) ).
% Int_atLeastAtMostL1
thf(fact_755_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_756_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_757_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_758_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_759_insert__subset,axiom,
! [X3: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( ( member_a @ X3 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_760_insert__subset,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X3 @ A ) @ B )
= ( ( member_set_a @ X3 @ B )
& ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_761_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_762_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_763_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_764_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_765_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_766_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_767_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_768_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_769_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_770_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_771_Un__insert__right,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( sup_sup_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_772_Un__insert__right,axiom,
! [A: set_set_a,A2: set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_773_Un__insert__left,axiom,
! [A2: a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_774_Un__insert__left,axiom,
! [A2: set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_775_max_Oabsorb1,axiom,
! [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_max_a @ A2 @ B2 )
= A2 ) ) ).
% max.absorb1
thf(fact_776_max_Oabsorb2,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_max_a @ A2 @ B2 )
= B2 ) ) ).
% max.absorb2
thf(fact_777_max_Obounded__iff,axiom,
! [B2: a,C: a,A2: a] :
( ( ord_less_eq_a @ ( ord_max_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_a @ B2 @ A2 )
& ( ord_less_eq_a @ C @ A2 ) ) ) ).
% max.bounded_iff
thf(fact_778_min_Oabsorb1,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_min_a @ A2 @ B2 )
= A2 ) ) ).
% min.absorb1
thf(fact_779_min_Oabsorb2,axiom,
! [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_min_a @ A2 @ B2 )
= B2 ) ) ).
% min.absorb2
thf(fact_780_min_Obounded__iff,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ ( ord_min_a @ B2 @ C ) )
= ( ( ord_less_eq_a @ A2 @ B2 )
& ( ord_less_eq_a @ A2 @ C ) ) ) ).
% min.bounded_iff
thf(fact_781_max__bot,axiom,
! [X3: set_a] :
( ( ord_max_set_a @ bot_bot_set_a @ X3 )
= X3 ) ).
% max_bot
thf(fact_782_max__bot2,axiom,
! [X3: set_a] :
( ( ord_max_set_a @ X3 @ bot_bot_set_a )
= X3 ) ).
% max_bot2
thf(fact_783_min__bot,axiom,
! [X3: set_a] :
( ( ord_min_set_a @ bot_bot_set_a @ X3 )
= bot_bot_set_a ) ).
% min_bot
thf(fact_784_min__bot2,axiom,
! [X3: set_a] :
( ( ord_min_set_a @ X3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% min_bot2
thf(fact_785_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_786_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B2: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_atLeastAtMost__singleton__iff,axiom,
! [A2: a,B2: a,C: a] :
( ( ( set_or672772299803893939Most_a @ A2 @ B2 )
= ( insert_a @ C @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( B2 = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_796_atLeastAtMost__singleton,axiom,
! [A2: a] :
( ( set_or672772299803893939Most_a @ A2 @ A2 )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% atLeastAtMost_singleton
thf(fact_797_Int__atMost,axiom,
! [A2: a,B2: a] :
( ( inf_inf_set_a @ ( set_ord_atMost_a @ A2 ) @ ( set_ord_atMost_a @ B2 ) )
= ( set_ord_atMost_a @ ( ord_min_a @ A2 @ B2 ) ) ) ).
% Int_atMost
thf(fact_798_Int__atLeastAtMostR1,axiom,
! [B2: a,C: a,D2: a] :
( ( inf_inf_set_a @ ( set_ord_atMost_a @ B2 ) @ ( set_or672772299803893939Most_a @ C @ D2 ) )
= ( set_or672772299803893939Most_a @ C @ ( ord_min_a @ B2 @ D2 ) ) ) ).
% Int_atLeastAtMostR1
thf(fact_799_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B7: set_a] :
( ( A
= ( insert_a @ A2 @ B7 ) )
& ~ ( member_a @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_800_mk__disjoint__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ? [B7: set_set_a] :
( ( A
= ( insert_set_a @ A2 @ B7 ) )
& ~ ( member_set_a @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_801_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_802_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_803_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_804_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_805_insert__ident,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ~ ( member_a @ X3 @ B )
=> ( ( ( insert_a @ X3 @ A )
= ( insert_a @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_806_insert__ident,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ~ ( member_set_a @ X3 @ B )
=> ( ( ( insert_set_a @ X3 @ A )
= ( insert_set_a @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_807_Set_Oset__insert,axiom,
! [X3: a,A: set_a] :
( ( member_a @ X3 @ A )
=> ~ ! [B7: set_a] :
( ( A
= ( insert_a @ X3 @ B7 ) )
=> ( member_a @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_808_Set_Oset__insert,axiom,
! [X3: set_a,A: set_set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ! [B7: set_set_a] :
( ( A
= ( insert_set_a @ X3 @ B7 ) )
=> ( member_set_a @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_809_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_810_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_811_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_812_insertI1,axiom,
! [A2: set_a,B: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B ) ) ).
% insertI1
thf(fact_813_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_814_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_815_max_Omono,axiom,
! [C: a,A2: a,D2: a,B2: a] :
( ( ord_less_eq_a @ C @ A2 )
=> ( ( ord_less_eq_a @ D2 @ B2 )
=> ( ord_less_eq_a @ ( ord_max_a @ C @ D2 ) @ ( ord_max_a @ A2 @ B2 ) ) ) ) ).
% max.mono
thf(fact_816_max_OorderE,axiom,
! [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( A2
= ( ord_max_a @ A2 @ B2 ) ) ) ).
% max.orderE
thf(fact_817_max_OorderI,axiom,
! [A2: a,B2: a] :
( ( A2
= ( ord_max_a @ A2 @ B2 ) )
=> ( ord_less_eq_a @ B2 @ A2 ) ) ).
% max.orderI
thf(fact_818_max_OboundedE,axiom,
! [B2: a,C: a,A2: a] :
( ( ord_less_eq_a @ ( ord_max_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_a @ B2 @ A2 )
=> ~ ( ord_less_eq_a @ C @ A2 ) ) ) ).
% max.boundedE
thf(fact_819_max_OboundedI,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ C @ A2 )
=> ( ord_less_eq_a @ ( ord_max_a @ B2 @ C ) @ A2 ) ) ) ).
% max.boundedI
thf(fact_820_max_Oorder__iff,axiom,
( ord_less_eq_a
= ( ^ [B5: a,A5: a] :
( A5
= ( ord_max_a @ A5 @ B5 ) ) ) ) ).
% max.order_iff
thf(fact_821_max_Ocobounded1,axiom,
! [A2: a,B2: a] : ( ord_less_eq_a @ A2 @ ( ord_max_a @ A2 @ B2 ) ) ).
% max.cobounded1
thf(fact_822_max_Ocobounded2,axiom,
! [B2: a,A2: a] : ( ord_less_eq_a @ B2 @ ( ord_max_a @ A2 @ B2 ) ) ).
% max.cobounded2
thf(fact_823_le__max__iff__disj,axiom,
! [Z: a,X3: a,Y2: a] :
( ( ord_less_eq_a @ Z @ ( ord_max_a @ X3 @ Y2 ) )
= ( ( ord_less_eq_a @ Z @ X3 )
| ( ord_less_eq_a @ Z @ Y2 ) ) ) ).
% le_max_iff_disj
thf(fact_824_max_Oabsorb__iff1,axiom,
( ord_less_eq_a
= ( ^ [B5: a,A5: a] :
( ( ord_max_a @ A5 @ B5 )
= A5 ) ) ) ).
% max.absorb_iff1
thf(fact_825_max_Oabsorb__iff2,axiom,
( ord_less_eq_a
= ( ^ [A5: a,B5: a] :
( ( ord_max_a @ A5 @ B5 )
= B5 ) ) ) ).
% max.absorb_iff2
thf(fact_826_max_OcoboundedI1,axiom,
! [C: a,A2: a,B2: a] :
( ( ord_less_eq_a @ C @ A2 )
=> ( ord_less_eq_a @ C @ ( ord_max_a @ A2 @ B2 ) ) ) ).
% max.coboundedI1
thf(fact_827_max_OcoboundedI2,axiom,
! [C: a,B2: a,A2: a] :
( ( ord_less_eq_a @ C @ B2 )
=> ( ord_less_eq_a @ C @ ( ord_max_a @ A2 @ B2 ) ) ) ).
% max.coboundedI2
thf(fact_828_max__def,axiom,
( ord_max_a
= ( ^ [A5: a,B5: a] : ( if_a @ ( ord_less_eq_a @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% max_def
thf(fact_829_max__absorb1,axiom,
! [Y2: a,X3: a] :
( ( ord_less_eq_a @ Y2 @ X3 )
=> ( ( ord_max_a @ X3 @ Y2 )
= X3 ) ) ).
% max_absorb1
thf(fact_830_max__absorb2,axiom,
! [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( ord_max_a @ X3 @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_831_min_Omono,axiom,
! [A2: a,C: a,B2: a,D2: a] :
( ( ord_less_eq_a @ A2 @ C )
=> ( ( ord_less_eq_a @ B2 @ D2 )
=> ( ord_less_eq_a @ ( ord_min_a @ A2 @ B2 ) @ ( ord_min_a @ C @ D2 ) ) ) ) ).
% min.mono
thf(fact_832_min_OorderE,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( A2
= ( ord_min_a @ A2 @ B2 ) ) ) ).
% min.orderE
thf(fact_833_min_OorderI,axiom,
! [A2: a,B2: a] :
( ( A2
= ( ord_min_a @ A2 @ B2 ) )
=> ( ord_less_eq_a @ A2 @ B2 ) ) ).
% min.orderI
thf(fact_834_min_OboundedE,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ ( ord_min_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_a @ A2 @ B2 )
=> ~ ( ord_less_eq_a @ A2 @ C ) ) ) ).
% min.boundedE
thf(fact_835_min_OboundedI,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ A2 @ C )
=> ( ord_less_eq_a @ A2 @ ( ord_min_a @ B2 @ C ) ) ) ) ).
% min.boundedI
thf(fact_836_min_Oorder__iff,axiom,
( ord_less_eq_a
= ( ^ [A5: a,B5: a] :
( A5
= ( ord_min_a @ A5 @ B5 ) ) ) ) ).
% min.order_iff
thf(fact_837_min_Ocobounded1,axiom,
! [A2: a,B2: a] : ( ord_less_eq_a @ ( ord_min_a @ A2 @ B2 ) @ A2 ) ).
% min.cobounded1
thf(fact_838_min_Ocobounded2,axiom,
! [A2: a,B2: a] : ( ord_less_eq_a @ ( ord_min_a @ A2 @ B2 ) @ B2 ) ).
% min.cobounded2
thf(fact_839_min_Oabsorb__iff1,axiom,
( ord_less_eq_a
= ( ^ [A5: a,B5: a] :
( ( ord_min_a @ A5 @ B5 )
= A5 ) ) ) ).
% min.absorb_iff1
thf(fact_840_min_Oabsorb__iff2,axiom,
( ord_less_eq_a
= ( ^ [B5: a,A5: a] :
( ( ord_min_a @ A5 @ B5 )
= B5 ) ) ) ).
% min.absorb_iff2
thf(fact_841_min_OcoboundedI1,axiom,
! [A2: a,C: a,B2: a] :
( ( ord_less_eq_a @ A2 @ C )
=> ( ord_less_eq_a @ ( ord_min_a @ A2 @ B2 ) @ C ) ) ).
% min.coboundedI1
thf(fact_842_min_OcoboundedI2,axiom,
! [B2: a,C: a,A2: a] :
( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_eq_a @ ( ord_min_a @ A2 @ B2 ) @ C ) ) ).
% min.coboundedI2
thf(fact_843_min__le__iff__disj,axiom,
! [X3: a,Y2: a,Z: a] :
( ( ord_less_eq_a @ ( ord_min_a @ X3 @ Y2 ) @ Z )
= ( ( ord_less_eq_a @ X3 @ Z )
| ( ord_less_eq_a @ Y2 @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_844_min__def,axiom,
( ord_min_a
= ( ^ [A5: a,B5: a] : ( if_a @ ( ord_less_eq_a @ A5 @ B5 ) @ A5 @ B5 ) ) ) ).
% min_def
thf(fact_845_min__absorb1,axiom,
! [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( ord_min_a @ X3 @ Y2 )
= X3 ) ) ).
% min_absorb1
thf(fact_846_min__absorb2,axiom,
! [Y2: a,X3: a] :
( ( ord_less_eq_a @ Y2 @ X3 )
=> ( ( ord_min_a @ X3 @ Y2 )
= Y2 ) ) ).
% min_absorb2
thf(fact_847_sets_Ogenerated__ringI__Basic,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ A @ ( sigma_6990475660951004707ring_a @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.generated_ringI_Basic
thf(fact_848_insert__subsetI,axiom,
! [X3: a,A: set_a,X4: set_a] :
( ( member_a @ X3 @ A )
=> ( ( ord_less_eq_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X3 @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_849_insert__subsetI,axiom,
! [X3: set_a,A: set_set_a,X4: set_set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ A )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X3 @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_850_subset__insert,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_851_subset__insert,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X3 @ B ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_852_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_853_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_854_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_855_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_856_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_857_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_858_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_859_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_860_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_861_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_862_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_863_sets_Ogenerated__ring__empty,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_6990475660951004707ring_a @ ( sigma_sets_a @ M ) ) ) ).
% sets.generated_ring_empty
thf(fact_864_sets_Ogenerated__ring__Int,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_6990475660951004707ring_a @ ( sigma_sets_a @ M ) ) )
=> ( ( member_set_a @ B2 @ ( sigma_6990475660951004707ring_a @ ( sigma_sets_a @ M ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( sigma_6990475660951004707ring_a @ ( sigma_sets_a @ M ) ) ) ) ) ).
% sets.generated_ring_Int
thf(fact_865_subset__singletonD,axiom,
! [A: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_866_subset__singleton__iff,axiom,
! [X4: set_a,A2: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_867_sets_Oinsert__in__sets,axiom,
! [X3: a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( insert_a @ X3 @ A ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.insert_in_sets
thf(fact_868_atLeastAtMost__singleton_H,axiom,
! [A2: a,B2: a] :
( ( A2 = B2 )
=> ( ( set_or672772299803893939Most_a @ A2 @ B2 )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_869_insert__is__Un,axiom,
( insert_set_a
= ( ^ [A5: set_a] : ( sup_sup_set_set_a @ ( insert_set_a @ A5 @ bot_bot_set_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_870_insert__is__Un,axiom,
( insert_a
= ( ^ [A5: a] : ( sup_sup_set_a @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_871_Un__singleton__iff,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a] :
( ( ( sup_sup_set_set_a @ A @ B )
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
= ( ( ( A = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) )
| ( ( A
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_872_Un__singleton__iff,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ( sup_sup_set_a @ A @ B )
= ( insert_a @ X3 @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X3 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_873_singleton__Un__iff,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ( ( insert_set_a @ X3 @ bot_bot_set_set_a )
= ( sup_sup_set_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) )
| ( ( A
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_874_singleton__Un__iff,axiom,
! [X3: a,A: set_a,B: set_a] :
( ( ( insert_a @ X3 @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X3 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_875_borel__singleton,axiom,
! [A: set_a,X3: a] :
( ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
=> ( member_set_a @ ( insert_a @ X3 @ A ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ).
% borel_singleton
thf(fact_876_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_877_the__elem__eq,axiom,
! [X3: a] :
( ( the_elem_a @ ( insert_a @ X3 @ bot_bot_set_a ) )
= X3 ) ).
% the_elem_eq
thf(fact_878_is__singletonI,axiom,
! [X3: a] : ( is_singleton_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_879_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
( A3
= ( insert_a @ ( the_elem_a @ A3 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_880_is__singletonI_H,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y3: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( member_set_a @ Y3 @ A )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_set_a @ A ) ) ) ).
% is_singletonI'
thf(fact_881_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y3 @ A )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_882_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_883_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_884_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
? [X: a] :
( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_885_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X2: a] :
( A
!= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_886_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_887_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_888_increasingD,axiom,
! [M: set_set_a,F: set_a > a,X3: set_a,Y2: set_a] :
( ( measur632008712001695539ng_a_a @ M @ F )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( member_set_a @ X3 @ M )
=> ( ( member_set_a @ Y2 @ M )
=> ( ord_less_eq_a @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_889_ivl__disj__un__singleton_I5_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( insert_a @ L @ bot_bot_set_a ) @ ( set_or4472690218693186638Most_a @ L @ U ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_890_semiring__of__sets_Ogenerated__ring__disjoint__Un,axiom,
! [Omega: set_set_a,M: set_set_set_a,A2: set_set_a,B2: set_set_a] :
( ( sigma_6809455824285593704_set_a @ Omega @ M )
=> ( ( member_set_set_a @ A2 @ ( sigma_7997604194779675523_set_a @ M ) )
=> ( ( member_set_set_a @ B2 @ ( sigma_7997604194779675523_set_a @ M ) )
=> ( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( sigma_7997604194779675523_set_a @ M ) ) ) ) ) ) ).
% semiring_of_sets.generated_ring_disjoint_Un
thf(fact_891_semiring__of__sets_Ogenerated__ring__disjoint__Un,axiom,
! [Omega: set_a,M: set_set_a,A2: set_a,B2: set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( ( member_set_a @ A2 @ ( sigma_6990475660951004707ring_a @ M ) )
=> ( ( member_set_a @ B2 @ ( sigma_6990475660951004707ring_a @ M ) )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sigma_6990475660951004707ring_a @ M ) ) ) ) ) ) ).
% semiring_of_sets.generated_ring_disjoint_Un
thf(fact_892_fmeasurable_Ogenerated__ring__disjoint__Un,axiom,
! [A2: set_set_a,M: sigma_measure_set_a,B2: set_set_a] :
( ( member_set_set_a @ A2 @ ( sigma_7997604194779675523_set_a @ ( measur7460903245211743562_set_a @ M ) ) )
=> ( ( member_set_set_a @ B2 @ ( sigma_7997604194779675523_set_a @ ( measur7460903245211743562_set_a @ M ) ) )
=> ( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( sigma_7997604194779675523_set_a @ ( measur7460903245211743562_set_a @ M ) ) ) ) ) ) ).
% fmeasurable.generated_ring_disjoint_Un
thf(fact_893_fmeasurable_Ogenerated__ring__disjoint__Un,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_6990475660951004707ring_a @ ( measur3645360004775918570able_a @ M ) ) )
=> ( ( member_set_a @ B2 @ ( sigma_6990475660951004707ring_a @ ( measur3645360004775918570able_a @ M ) ) )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sigma_6990475660951004707ring_a @ ( measur3645360004775918570able_a @ M ) ) ) ) ) ) ).
% fmeasurable.generated_ring_disjoint_Un
thf(fact_894_fmeasurable_Oempty__sets,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( measur3645360004775918570able_a @ M ) ) ).
% fmeasurable.empty_sets
thf(fact_895_fmeasurable_OInt,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ B2 @ ( measur3645360004775918570able_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable.Int
thf(fact_896_fmeasurable_OUn,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ B2 @ ( measur3645360004775918570able_a @ M ) )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable.Un
thf(fact_897_fmeasurable_OUn,axiom,
! [A2: set_set_a,M: sigma_measure_set_a,B2: set_set_a] :
( ( member_set_set_a @ A2 @ ( measur7460903245211743562_set_a @ M ) )
=> ( ( member_set_set_a @ B2 @ ( measur7460903245211743562_set_a @ M ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( measur7460903245211743562_set_a @ M ) ) ) ) ).
% fmeasurable.Un
thf(fact_898_greaterThanAtMost__empty,axiom,
! [L: a,K: a] :
( ( ord_less_eq_a @ L @ K )
=> ( ( set_or4472690218693186638Most_a @ K @ L )
= bot_bot_set_a ) ) ).
% greaterThanAtMost_empty
thf(fact_899_fmeasurable_Ogenerated__ringI__Basic,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M ) )
=> ( member_set_a @ A @ ( sigma_6990475660951004707ring_a @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable.generated_ringI_Basic
thf(fact_900_fmeasurableD,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M ) )
=> ( member_set_a @ A @ ( sigma_sets_a @ M ) ) ) ).
% fmeasurableD
thf(fact_901_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_902_semiring__of__sets_Oempty__sets,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( member_set_a @ bot_bot_set_a @ M ) ) ).
% semiring_of_sets.empty_sets
thf(fact_903_semiring__of__sets_OInt__space__eq2,axiom,
! [Omega: set_a,M: set_set_a,X3: set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( ( member_set_a @ X3 @ M )
=> ( ( inf_inf_set_a @ X3 @ Omega )
= X3 ) ) ) ).
% semiring_of_sets.Int_space_eq2
thf(fact_904_semiring__of__sets_OInt__space__eq1,axiom,
! [Omega: set_a,M: set_set_a,X3: set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( ( member_set_a @ X3 @ M )
=> ( ( inf_inf_set_a @ Omega @ X3 )
= X3 ) ) ) ).
% semiring_of_sets.Int_space_eq1
thf(fact_905_semiring__of__sets_OInt,axiom,
! [Omega: set_a,M: set_set_a,A2: set_a,B2: set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( ( member_set_a @ A2 @ M )
=> ( ( member_set_a @ B2 @ M )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ M ) ) ) ) ).
% semiring_of_sets.Int
thf(fact_906_semiring__of__sets_Ogenerated__ringI__Basic,axiom,
! [Omega: set_a,M: set_set_a,A: set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( ( member_set_a @ A @ M )
=> ( member_set_a @ A @ ( sigma_6990475660951004707ring_a @ M ) ) ) ) ).
% semiring_of_sets.generated_ringI_Basic
thf(fact_907_fmeasurable_Oinsert__in__sets,axiom,
! [X3: a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M ) )
=> ( member_set_a @ ( insert_a @ X3 @ A ) @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable.insert_in_sets
thf(fact_908_fmeasurableI2,axiom,
! [A: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ B @ ( measur3645360004775918570able_a @ M ) ) ) ) ) ).
% fmeasurableI2
thf(fact_909_fmeasurable__Int__fmeasurable,axiom,
! [S: set_a,M: sigma_measure_a,T2: set_a] :
( ( member_set_a @ S @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ T2 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ S @ T2 ) @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable_Int_fmeasurable
thf(fact_910_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_911_fmeasurable_Ogenerated__ring__empty,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_6990475660951004707ring_a @ ( measur3645360004775918570able_a @ M ) ) ) ).
% fmeasurable.generated_ring_empty
thf(fact_912_ivl__disj__un__two_I6_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or4472690218693186638Most_a @ M2 @ U ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_913_fmeasurable_Ogenerated__ring__Int,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_6990475660951004707ring_a @ ( measur3645360004775918570able_a @ M ) ) )
=> ( ( member_set_a @ B2 @ ( sigma_6990475660951004707ring_a @ ( measur3645360004775918570able_a @ M ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( sigma_6990475660951004707ring_a @ ( measur3645360004775918570able_a @ M ) ) ) ) ) ).
% fmeasurable.generated_ring_Int
thf(fact_914_ivl__disj__int__two_I6_J,axiom,
! [L: a,M2: a,U: a] :
( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or4472690218693186638Most_a @ M2 @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(6)
thf(fact_915_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_916_semiring__of__sets_Ogenerated__ring__empty,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( member_set_a @ bot_bot_set_a @ ( sigma_6990475660951004707ring_a @ M ) ) ) ).
% semiring_of_sets.generated_ring_empty
thf(fact_917_semiring__of__sets_Ogenerated__ring__Int,axiom,
! [Omega: set_a,M: set_set_a,A2: set_a,B2: set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( ( member_set_a @ A2 @ ( sigma_6990475660951004707ring_a @ M ) )
=> ( ( member_set_a @ B2 @ ( sigma_6990475660951004707ring_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( sigma_6990475660951004707ring_a @ M ) ) ) ) ) ).
% semiring_of_sets.generated_ring_Int
thf(fact_918_Ioc__disjoint,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ A2 @ B2 ) @ ( set_or4472690218693186638Most_a @ C @ D2 ) )
= bot_bot_set_a )
= ( ( ord_less_eq_a @ B2 @ A2 )
| ( ord_less_eq_a @ D2 @ C )
| ( ord_less_eq_a @ B2 @ C )
| ( ord_less_eq_a @ D2 @ A2 ) ) ) ).
% Ioc_disjoint
thf(fact_919_ivl__disj__un__two_I8_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or4472690218693186638Most_a @ M2 @ U ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_920_ivl__disj__int__two_I8_J,axiom,
! [L: a,M2: a,U: a] :
( ( inf_inf_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or4472690218693186638Most_a @ M2 @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(8)
thf(fact_921_ivl__disj__un__one_I3_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or4472690218693186638Most_a @ L @ U ) )
= ( set_ord_atMost_a @ U ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_922_ivl__disj__int__one_I3_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or4472690218693186638Most_a @ L @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(3)
thf(fact_923_ivl__disj__un__one_I5_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ U ) @ ( set_or8632414552788122084Than_a @ U ) )
= ( set_or8632414552788122084Than_a @ L ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_924_ivl__disj__int__one_I5_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ L @ U ) @ ( set_or8632414552788122084Than_a @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(5)
thf(fact_925_greaterThanAtMost__def,axiom,
( set_or4472690218693186638Most_a
= ( ^ [L3: a,U2: a] : ( inf_inf_set_a @ ( set_or8632414552788122084Than_a @ L3 ) @ ( set_ord_atMost_a @ U2 ) ) ) ) ).
% greaterThanAtMost_def
thf(fact_926_null__sets_Ogenerated__ring__disjoint__Un,axiom,
! [A2: set_set_a,M: sigma_measure_set_a,B2: set_set_a] :
( ( member_set_set_a @ A2 @ ( sigma_7997604194779675523_set_a @ ( measur1516554128032400785_set_a @ M ) ) )
=> ( ( member_set_set_a @ B2 @ ( sigma_7997604194779675523_set_a @ ( measur1516554128032400785_set_a @ M ) ) )
=> ( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( sigma_7997604194779675523_set_a @ ( measur1516554128032400785_set_a @ M ) ) ) ) ) ) ).
% null_sets.generated_ring_disjoint_Un
thf(fact_927_null__sets_Ogenerated__ring__disjoint__Un,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_6990475660951004707ring_a @ ( measure_null_sets_a @ M ) ) )
=> ( ( member_set_a @ B2 @ ( sigma_6990475660951004707ring_a @ ( measure_null_sets_a @ M ) ) )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sigma_6990475660951004707ring_a @ ( measure_null_sets_a @ M ) ) ) ) ) ) ).
% null_sets.generated_ring_disjoint_Un
thf(fact_928_ivl__disj__un__singleton_I6_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ U ) @ ( insert_a @ U @ bot_bot_set_a ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_929_space__empty__iff,axiom,
! [N2: sigma_measure_a] :
( ( ( sigma_space_a @ N2 )
= bot_bot_set_a )
= ( ( sigma_sets_a @ N2 )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ) ).
% space_empty_iff
thf(fact_930_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
( set_or4472690218693186638Most_a
= ( ^ [A5: a,B5: a] : ( minus_minus_set_a @ ( set_or672772299803893939Most_a @ A5 @ B5 ) @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_931_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_932_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_933_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_934_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_935_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_936_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_937_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_938_sets_Otop,axiom,
! [M: sigma_measure_a] : ( member_set_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.top
thf(fact_939_Diff__insert0,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_940_Diff__insert0,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X3 @ B ) )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_941_insert__Diff1,axiom,
! [X3: a,B: set_a,A: set_a] :
( ( member_a @ X3 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_942_insert__Diff1,axiom,
! [X3: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ X3 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X3 @ A ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_943_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_944_Un__Diff__cancel,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_945_Un__Diff__cancel,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ A ) )
= ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_946_Un__Diff__cancel2,axiom,
! [B: set_a,A: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A ) @ A )
= ( sup_sup_set_a @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_947_Un__Diff__cancel2,axiom,
! [B: set_set_a,A: set_set_a] :
( ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ B @ A ) @ A )
= ( sup_sup_set_set_a @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_948_null__sets_Oempty__sets,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( measure_null_sets_a @ M ) ) ).
% null_sets.empty_sets
thf(fact_949_null__sets_OInt,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ B2 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets.Int
thf(fact_950_null__sets_OUn,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ B2 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets.Un
thf(fact_951_null__sets_OUn,axiom,
! [A2: set_set_a,M: sigma_measure_set_a,B2: set_set_a] :
( ( member_set_set_a @ A2 @ ( measur1516554128032400785_set_a @ M ) )
=> ( ( member_set_set_a @ B2 @ ( measur1516554128032400785_set_a @ M ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( measur1516554128032400785_set_a @ M ) ) ) ) ).
% null_sets.Un
thf(fact_952_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_953_ivl__subset,axiom,
! [I2: a,J: a,M2: a,N: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ I2 @ J ) @ ( set_or5139330845457685135Than_a @ M2 @ N ) )
= ( ( ord_less_eq_a @ J @ I2 )
| ( ( ord_less_eq_a @ M2 @ I2 )
& ( ord_less_eq_a @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_954_atLeastLessThan__empty,axiom,
! [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_955_insert__Diff__single,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( insert_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_956_Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_957_sets_OInt__space__eq2,axiom,
! [X3: set_a,M: sigma_measure_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ X3 @ ( sigma_space_a @ M ) )
= X3 ) ) ).
% sets.Int_space_eq2
thf(fact_958_sets_OInt__space__eq1,axiom,
! [X3: set_a,M: sigma_measure_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M ) @ X3 )
= X3 ) ) ).
% sets.Int_space_eq1
thf(fact_959_ivl__diff,axiom,
! [I2: a,N: a,M2: a] :
( ( ord_less_eq_a @ I2 @ N )
=> ( ( minus_minus_set_a @ ( set_or5139330845457685135Than_a @ I2 @ M2 ) @ ( set_or5139330845457685135Than_a @ I2 @ N ) )
= ( set_or5139330845457685135Than_a @ N @ M2 ) ) ) ).
% ivl_diff
thf(fact_960_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_961_fmeasurable_OInt__space__eq1,axiom,
! [X3: set_a,M: sigma_measure_a] :
( ( member_set_a @ X3 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M ) @ X3 )
= X3 ) ) ).
% fmeasurable.Int_space_eq1
thf(fact_962_fmeasurable_OInt__space__eq2,axiom,
! [X3: set_a,M: sigma_measure_a] :
( ( member_set_a @ X3 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( inf_inf_set_a @ X3 @ ( sigma_space_a @ M ) )
= X3 ) ) ).
% fmeasurable.Int_space_eq2
thf(fact_963_null__sets_OInt__space__eq1,axiom,
! [X3: set_a,M: sigma_measure_a] :
( ( member_set_a @ X3 @ ( measure_null_sets_a @ M ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M ) @ X3 )
= X3 ) ) ).
% null_sets.Int_space_eq1
thf(fact_964_null__sets_OInt__space__eq2,axiom,
! [X3: set_a,M: sigma_measure_a] :
( ( member_set_a @ X3 @ ( measure_null_sets_a @ M ) )
=> ( ( inf_inf_set_a @ X3 @ ( sigma_space_a @ M ) )
= X3 ) ) ).
% null_sets.Int_space_eq2
thf(fact_965_space__bot,axiom,
( ( sigma_space_a @ bot_bo2108912051383640591sure_a )
= bot_bot_set_a ) ).
% space_bot
thf(fact_966_single__Diff__lessThan,axiom,
! [K: a] :
( ( minus_minus_set_a @ ( insert_a @ K @ bot_bot_set_a ) @ ( set_ord_lessThan_a @ K ) )
= ( insert_a @ K @ bot_bot_set_a ) ) ).
% single_Diff_lessThan
thf(fact_967_null__sets_Osets__into__space,axiom,
! [X3: set_a,M: sigma_measure_a] :
( ( member_set_a @ X3 @ ( measure_null_sets_a @ M ) )
=> ( ord_less_eq_set_a @ X3 @ ( sigma_space_a @ M ) ) ) ).
% null_sets.sets_into_space
thf(fact_968_measurable__Diff__null__set,axiom,
! [B: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( ( member_set_a @ ( minus_minus_set_a @ A @ B ) @ ( measur3645360004775918570able_a @ M ) )
& ( member_set_a @ A @ ( sigma_sets_a @ M ) ) )
= ( member_set_a @ A @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% measurable_Diff_null_set
thf(fact_969_Un__Diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ C2 ) @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_970_Un__Diff,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ A @ C2 ) @ ( minus_5736297505244876581_set_a @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_971_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_972_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_973_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_974_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_975_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_976_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_977_insert__Diff__if,axiom,
! [X3: a,B: set_a,A: set_a] :
( ( ( member_a @ X3 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X3 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( insert_a @ X3 @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_978_insert__Diff__if,axiom,
! [X3: set_a,B: set_set_a,A: set_set_a] :
( ( ( member_set_a @ X3 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X3 @ A ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) )
& ( ~ ( member_set_a @ X3 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X3 @ A ) @ B )
= ( insert_set_a @ X3 @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_979_sets__eq__imp__space__eq,axiom,
! [M: sigma_measure_a,M3: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M3 ) )
=> ( ( sigma_space_a @ M )
= ( sigma_space_a @ M3 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_980_null__setsD2,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ A @ ( sigma_sets_a @ M ) ) ) ).
% null_setsD2
thf(fact_981_null__set__Diff,axiom,
! [B: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ B @ A ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_set_Diff
thf(fact_982_null__sets_Ogenerated__ringI__Basic,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ A @ ( sigma_6990475660951004707ring_a @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets.generated_ringI_Basic
thf(fact_983_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or5139330845457685135Than_a
= ( ^ [A5: a,B5: a] : ( minus_minus_set_a @ ( set_or672772299803893939Most_a @ A5 @ B5 ) @ ( insert_a @ B5 @ bot_bot_set_a ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_984_closed__cdi__Compl,axiom,
! [Omega: set_a,M: set_set_a,S2: set_a] :
( ( sigma_closed_cdi_a @ Omega @ M )
=> ( ( member_set_a @ S2 @ M )
=> ( member_set_a @ ( minus_minus_set_a @ Omega @ S2 ) @ M ) ) ) ).
% closed_cdi_Compl
thf(fact_985_atLeastLessThan__subset__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ A2 @ B2 ) @ ( set_or5139330845457685135Than_a @ C @ D2 ) )
=> ( ( ord_less_eq_a @ B2 @ A2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_986_ivl__disj__un__two_I3_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_987_ivl__disj__int__two_I3_J,axiom,
! [L: a,M2: a,U: a] :
( ( inf_inf_set_a @ ( set_or5139330845457685135Than_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(3)
thf(fact_988_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_989_null__sets_Oinsert__in__sets,axiom,
! [X3: a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ ( insert_a @ X3 @ A ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets.insert_in_sets
thf(fact_990_null__set__Int2,axiom,
! [B: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ B @ A ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_set_Int2
thf(fact_991_null__set__Int1,axiom,
! [B: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A @ B ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_set_Int1
thf(fact_992_diff__shunt__var,axiom,
! [X3: set_a,Y2: set_a] :
( ( ( minus_minus_set_a @ X3 @ Y2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X3 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_993_sets_Osets__into__space,axiom,
! [X3: set_a,M: sigma_measure_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ M ) )
=> ( ord_less_eq_set_a @ X3 @ ( sigma_space_a @ M ) ) ) ).
% sets.sets_into_space
thf(fact_994_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_995_null__sets_Ogenerated__ring__empty,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_6990475660951004707ring_a @ ( measure_null_sets_a @ M ) ) ) ).
% null_sets.generated_ring_empty
thf(fact_996_null__sets_Ogenerated__ring__Int,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_6990475660951004707ring_a @ ( measure_null_sets_a @ M ) ) )
=> ( ( member_set_a @ B2 @ ( sigma_6990475660951004707ring_a @ ( measure_null_sets_a @ M ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( sigma_6990475660951004707ring_a @ ( measure_null_sets_a @ M ) ) ) ) ) ).
% null_sets.generated_ring_Int
thf(fact_997_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X3: a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_998_subset__Diff__insert,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ ( insert_set_a @ X3 @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ C2 ) )
& ~ ( member_set_a @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_999_Diff__insert__absorb,axiom,
! [X3: set_a,A: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X3 @ A ) @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1000_Diff__insert__absorb,axiom,
! [X3: a,A: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1001_Diff__insert2,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_1002_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_1003_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_1004_Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_1005_Int__Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_1006_Diff__triv,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1007_fmeasurable_Osets__into__space,axiom,
! [X3: set_a,M: sigma_measure_a] :
( ( member_set_a @ X3 @ ( measur3645360004775918570able_a @ M ) )
=> ( ord_less_eq_set_a @ X3 @ ( sigma_space_a @ M ) ) ) ).
% fmeasurable.sets_into_space
thf(fact_1008_Diff__subset__conv,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
= ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1009_Diff__subset__conv,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ C2 )
= ( ord_le3724670747650509150_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1010_Diff__partition,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1011_Diff__partition,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( sup_sup_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1012_Diff__Un,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1013_Diff__Un,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) )
= ( inf_inf_set_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1014_Diff__Int,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1015_Diff__Int,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) )
= ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1016_Int__Diff__Un,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1017_Int__Diff__Un,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1018_Un__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1019_Un__Diff__Int,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( inf_inf_set_set_a @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1020_fmeasurable__Diff__D,axiom,
! [T2: set_a,S: set_a,M: sigma_measure_a] :
( ( member_set_a @ ( minus_minus_set_a @ T2 @ S ) @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ S @ ( measur3645360004775918570able_a @ M ) )
=> ( ( ord_less_eq_set_a @ S @ T2 )
=> ( member_set_a @ T2 @ ( measur3645360004775918570able_a @ M ) ) ) ) ) ).
% fmeasurable_Diff_D
thf(fact_1021_fmeasurable__Diff,axiom,
! [A: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ A @ B ) @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable_Diff
thf(fact_1022_sets_Osemiring__of__sets__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_8461971822185508616sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.semiring_of_sets_axioms
thf(fact_1023_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_1024_space__empty__eq__bot,axiom,
! [A2: sigma_measure_a] :
( ( ( sigma_space_a @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bo2108912051383640591sure_a ) ) ).
% space_empty_eq_bot
thf(fact_1025_ivl__disj__un__two_I7_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_1026_ivl__disj__int__two_I7_J,axiom,
! [L: a,M2: a,U: a] :
( ( inf_inf_set_a @ ( set_or5139330845457685135Than_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(7)
thf(fact_1027_ivl__disj__un__one_I2_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or5139330845457685135Than_a @ L @ U ) )
= ( set_ord_lessThan_a @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_1028_ivl__disj__int__one_I2_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or5139330845457685135Than_a @ L @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(2)
thf(fact_1029_ivl__disj__un__one_I8_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= ( set_ord_atLeast_a @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_1030_ivl__disj__int__one_I8_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_or5139330845457685135Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(8)
thf(fact_1031_measurable__Un__null__set,axiom,
! [B: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( ( member_set_a @ ( sup_sup_set_a @ A @ B ) @ ( measur3645360004775918570able_a @ M ) )
& ( member_set_a @ A @ ( sigma_sets_a @ M ) ) )
= ( member_set_a @ A @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% measurable_Un_null_set
thf(fact_1032_measurable__Un__null__set,axiom,
! [B: set_set_a,M: sigma_measure_set_a,A: set_set_a] :
( ( member_set_set_a @ B @ ( measur1516554128032400785_set_a @ M ) )
=> ( ( ( member_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( measur7460903245211743562_set_a @ M ) )
& ( member_set_set_a @ A @ ( sigma_sets_set_a @ M ) ) )
= ( member_set_set_a @ A @ ( measur7460903245211743562_set_a @ M ) ) ) ) ).
% measurable_Un_null_set
thf(fact_1033_atLeastLessThan__def,axiom,
( set_or5139330845457685135Than_a
= ( ^ [L3: a,U2: a] : ( inf_inf_set_a @ ( set_ord_atLeast_a @ L3 ) @ ( set_ord_lessThan_a @ U2 ) ) ) ) ).
% atLeastLessThan_def
thf(fact_1034_Diff__single__insert,axiom,
! [A: set_a,X3: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1035_subset__insert__iff,axiom,
! [A: set_set_a,X3: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X3 @ B ) )
= ( ( ( member_set_a @ X3 @ A )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X3 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1036_subset__insert__iff,axiom,
! [A: set_a,X3: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( ( ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1037_null__sets__subset,axiom,
! [B: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_set_a @ A @ ( measure_null_sets_a @ M ) ) ) ) ) ).
% null_sets_subset
thf(fact_1038_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_1039_remove__def,axiom,
( remove_a
= ( ^ [X: a,A3: set_a] : ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_1040_algebra__single__set,axiom,
! [X4: set_a,S: set_a] :
( ( ord_less_eq_set_a @ X4 @ S )
=> ( sigma_algebra_a @ S @ ( insert_set_a @ bot_bot_set_a @ ( insert_set_a @ X4 @ ( insert_set_a @ ( minus_minus_set_a @ S @ X4 ) @ ( insert_set_a @ S @ bot_bot_set_set_a ) ) ) ) ) ) ).
% algebra_single_set
thf(fact_1041_member__remove,axiom,
! [X3: a,Y2: a,A: set_a] :
( ( member_a @ X3 @ ( remove_a @ Y2 @ A ) )
= ( ( member_a @ X3 @ A )
& ( X3 != Y2 ) ) ) ).
% member_remove
thf(fact_1042_member__remove,axiom,
! [X3: set_a,Y2: set_a,A: set_set_a] :
( ( member_set_a @ X3 @ ( remove_set_a @ Y2 @ A ) )
= ( ( member_set_a @ X3 @ A )
& ( X3 != Y2 ) ) ) ).
% member_remove
thf(fact_1043_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_1044_algebra_Ocompl__sets,axiom,
! [Omega: set_a,M: set_set_a,A2: set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( member_set_a @ A2 @ M )
=> ( member_set_a @ ( minus_minus_set_a @ Omega @ A2 ) @ M ) ) ) ).
% algebra.compl_sets
thf(fact_1045_algebra_Otop,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( member_set_a @ Omega @ M ) ) ).
% algebra.top
thf(fact_1046_sets_Oalgebra__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_algebra_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.algebra_axioms
thf(fact_1047_sets_Osmallest__ccdi__sets__Un,axiom,
! [A: set_set_a,M: sigma_measure_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( sigma_8136683933974138297_set_a @ ( sigma_space_set_a @ M ) @ ( sigma_sets_set_a @ M ) ) )
=> ( ( member_set_set_a @ B @ ( sigma_8136683933974138297_set_a @ ( sigma_space_set_a @ M ) @ ( sigma_sets_set_a @ M ) ) )
=> ( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sigma_8136683933974138297_set_a @ ( sigma_space_set_a @ M ) @ ( sigma_sets_set_a @ M ) ) ) ) ) ) ).
% sets.smallest_ccdi_sets_Un
thf(fact_1048_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_1049_sigma__algebra__single__set,axiom,
! [X4: set_a,S: set_a] :
( ( ord_less_eq_set_a @ X4 @ S )
=> ( sigma_4968961713055010667ebra_a @ S @ ( insert_set_a @ bot_bot_set_a @ ( insert_set_a @ X4 @ ( insert_set_a @ ( minus_minus_set_a @ S @ X4 ) @ ( insert_set_a @ S @ bot_bot_set_set_a ) ) ) ) ) ) ).
% sigma_algebra_single_set
thf(fact_1050_le__measure__iff,axiom,
( ord_le254669795585780187sure_a
= ( ^ [M4: sigma_measure_a,N3: sigma_measure_a] :
( ( ( ( sigma_space_a @ M4 )
= ( sigma_space_a @ N3 ) )
=> ( ( ( ( sigma_sets_a @ M4 )
= ( sigma_sets_a @ N3 ) )
=> ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ M4 ) @ ( sigma_emeasure_a @ N3 ) ) )
& ( ( ( sigma_sets_a @ M4 )
!= ( sigma_sets_a @ N3 ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M4 ) @ ( sigma_sets_a @ N3 ) ) ) ) )
& ( ( ( sigma_space_a @ M4 )
!= ( sigma_space_a @ N3 ) )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ M4 ) @ ( sigma_space_a @ N3 ) ) ) ) ) ) ).
% le_measure_iff
thf(fact_1051_measure__eqI,axiom,
! [M: sigma_measure_a,N2: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N2 ) )
=> ( ! [A7: set_a] :
( ( member_set_a @ A7 @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ A7 )
= ( sigma_emeasure_a @ N2 @ A7 ) ) )
=> ( M = N2 ) ) ) ).
% measure_eqI
thf(fact_1052_smallest__ccdi__sets_OBasic,axiom,
! [A2: set_a,M: set_set_a,Omega: set_a] :
( ( member_set_a @ A2 @ M )
=> ( member_set_a @ A2 @ ( sigma_5648178489087971417sets_a @ Omega @ M ) ) ) ).
% smallest_ccdi_sets.Basic
thf(fact_1053_smallest__ccdi__sets_OCompl,axiom,
! [A2: set_a,Omega: set_a,M: set_set_a] :
( ( member_set_a @ A2 @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ Omega @ A2 ) @ ( sigma_5648178489087971417sets_a @ Omega @ M ) ) ) ).
% smallest_ccdi_sets.Compl
thf(fact_1054_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_1055_sets_Osigma__algebra__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.sigma_algebra_axioms
thf(fact_1056_le__measure,axiom,
! [M: sigma_measure_a,N2: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N2 ) )
=> ( ( ord_le254669795585780187sure_a @ M @ N2 )
= ( ! [X: set_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ X ) @ ( sigma_emeasure_a @ N2 @ X ) ) ) ) ) ) ).
% le_measure
thf(fact_1057_le__measureD3,axiom,
! [A: sigma_measure_a,B: sigma_measure_a,X4: set_a] :
( ( ord_le254669795585780187sure_a @ A @ B )
=> ( ( ( sigma_sets_a @ A )
= ( sigma_sets_a @ B ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ A @ X4 ) @ ( sigma_emeasure_a @ B @ X4 ) ) ) ) ).
% le_measureD3
thf(fact_1058_algebra_Osmallest__ccdi__sets__Int1,axiom,
! [Omega: set_a,M: set_set_a,A2: set_a,B2: set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( member_set_a @ A2 @ M )
=> ( ( member_set_a @ B2 @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( sigma_5648178489087971417sets_a @ Omega @ M ) ) ) ) ) ).
% algebra.smallest_ccdi_sets_Int1
thf(fact_1059_algebra_Osmallest__ccdi__sets__Int,axiom,
! [Omega: set_a,M: set_set_a,B2: set_a,A2: set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( member_set_a @ B2 @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( ( member_set_a @ A2 @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( sigma_5648178489087971417sets_a @ Omega @ M ) ) ) ) ) ).
% algebra.smallest_ccdi_sets_Int
thf(fact_1060_le__emeasure__sup__measure_H1,axiom,
! [B: sigma_measure_a,A: sigma_measure_a,X4: set_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A ) )
=> ( ( member_set_a @ X4 @ ( sigma_sets_a @ A ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ A @ X4 ) @ ( sigma_emeasure_a @ ( measur3004909623614618064sure_a @ A @ B ) @ X4 ) ) ) ) ).
% le_emeasure_sup_measure'1
thf(fact_1061_le__emeasure__sup__measure_H2,axiom,
! [B: sigma_measure_a,A: sigma_measure_a,X4: set_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A ) )
=> ( ( member_set_a @ X4 @ ( sigma_sets_a @ A ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ B @ X4 ) @ ( sigma_emeasure_a @ ( measur3004909623614618064sure_a @ A @ B ) @ X4 ) ) ) ) ).
% le_emeasure_sup_measure'2
thf(fact_1062_emeasure__Diff__null__set,axiom,
! [B: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ ( minus_minus_set_a @ A @ B ) )
= ( sigma_emeasure_a @ M @ A ) ) ) ) ).
% emeasure_Diff_null_set
thf(fact_1063_emeasure__Un__null__set,axiom,
! [A: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ ( sup_sup_set_a @ A @ B ) )
= ( sigma_emeasure_a @ M @ A ) ) ) ) ).
% emeasure_Un_null_set
thf(fact_1064_emeasure__Un__null__set,axiom,
! [A: set_set_a,M: sigma_measure_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a @ B @ ( measur1516554128032400785_set_a @ M ) )
=> ( ( sigma_emeasure_set_a @ M @ ( sup_sup_set_set_a @ A @ B ) )
= ( sigma_emeasure_set_a @ M @ A ) ) ) ) ).
% emeasure_Un_null_set
thf(fact_1065_sets_Osmallest__ccdi__sets__Int1,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B2 @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ) ) ) ).
% sets.smallest_ccdi_sets_Int1
thf(fact_1066_sets_Osmallest__ccdi__sets__Int,axiom,
! [B2: set_a,M: sigma_measure_a,A2: set_a] :
( ( member_set_a @ B2 @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) )
=> ( ( member_set_a @ A2 @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ) ) ) ).
% sets.smallest_ccdi_sets_Int
thf(fact_1067_less__eq__measure_Ointros_I3_J,axiom,
! [M: sigma_measure_a,N2: sigma_measure_a] :
( ( ( sigma_space_a @ M )
= ( sigma_space_a @ N2 ) )
=> ( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N2 ) )
=> ( ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ M ) @ ( sigma_emeasure_a @ N2 ) )
=> ( ord_le254669795585780187sure_a @ M @ N2 ) ) ) ) ).
% less_eq_measure.intros(3)
thf(fact_1068_sets_Osmallest__closed__cdi1,axiom,
! [M: sigma_measure_a] : ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ) ).
% sets.smallest_closed_cdi1
thf(fact_1069_sigma__algebra__trivial,axiom,
! [Omega: set_a] : ( sigma_4968961713055010667ebra_a @ Omega @ ( insert_set_a @ bot_bot_set_a @ ( insert_set_a @ Omega @ bot_bot_set_set_a ) ) ) ).
% sigma_algebra_trivial
thf(fact_1070_emeasure__sup__measure_H__le2,axiom,
! [B: sigma_measure_a,C2: sigma_measure_a,A: sigma_measure_a,X4: set_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ C2 ) )
=> ( ( ( sigma_sets_a @ A )
= ( sigma_sets_a @ C2 ) )
=> ( ( member_set_a @ X4 @ ( sigma_sets_a @ C2 ) )
=> ( ! [Y6: set_a] :
( ( ord_less_eq_set_a @ Y6 @ X4 )
=> ( ( 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 @ X4 )
=> ( ( 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 ) @ X4 ) @ ( sigma_emeasure_a @ C2 @ X4 ) ) ) ) ) ) ) ).
% emeasure_sup_measure'_le2
thf(fact_1071_sets_Osmallest__closed__cdi2,axiom,
! [M: sigma_measure_a] : ( sigma_closed_cdi_a @ ( sigma_space_a @ M ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ) ).
% sets.smallest_closed_cdi2
thf(fact_1072_algebra_Osmallest__ccdi__sets__Un,axiom,
! [Omega: set_set_a,M: set_set_set_a,A: set_set_a,B: set_set_a] :
( ( sigma_algebra_set_a @ Omega @ M )
=> ( ( member_set_set_a @ A @ ( sigma_8136683933974138297_set_a @ Omega @ M ) )
=> ( ( member_set_set_a @ B @ ( sigma_8136683933974138297_set_a @ Omega @ M ) )
=> ( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sigma_8136683933974138297_set_a @ Omega @ M ) ) ) ) ) ) ).
% algebra.smallest_ccdi_sets_Un
thf(fact_1073_algebra_Osmallest__ccdi__sets__Un,axiom,
! [Omega: set_a,M: set_set_a,A: set_a,B: set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( member_set_a @ A @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( ( member_set_a @ B @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( member_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sigma_5648178489087971417sets_a @ Omega @ M ) ) ) ) ) ) ).
% algebra.smallest_ccdi_sets_Un
thf(fact_1074_outer__measure__of__attain,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ A @ ( sigma_space_a @ M ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( sigma_sets_a @ M ) )
& ( ord_less_eq_set_a @ A @ X2 )
& ( ( comple3326625653960060650e_of_a @ M @ A )
= ( sigma_emeasure_a @ M @ X2 ) ) ) ) ).
% outer_measure_of_attain
thf(fact_1075_outer__measure__of__eq,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( comple3326625653960060650e_of_a @ M @ A )
= ( sigma_emeasure_a @ M @ A ) ) ) ).
% outer_measure_of_eq
thf(fact_1076_measurable__envelope__def,axiom,
( comple1317578422686860819lope_a
= ( ^ [M4: sigma_measure_a,A3: set_a,E: set_a] :
( ( ord_less_eq_set_a @ A3 @ E )
& ( member_set_a @ E @ ( sigma_sets_a @ M4 ) )
& ! [X: set_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M4 ) )
=> ( ( sigma_emeasure_a @ M4 @ ( inf_inf_set_a @ X @ E ) )
= ( comple3326625653960060650e_of_a @ M4 @ ( inf_inf_set_a @ X @ A3 ) ) ) ) ) ) ) ).
% measurable_envelope_def
thf(fact_1077_measurable__envelopeD_I3_J,axiom,
! [M: sigma_measure_a,A: set_a,E2: set_a,F3: set_a] :
( ( comple1317578422686860819lope_a @ M @ A @ E2 )
=> ( ( member_set_a @ F3 @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ ( inf_inf_set_a @ F3 @ E2 ) )
= ( comple3326625653960060650e_of_a @ M @ ( inf_inf_set_a @ F3 @ A ) ) ) ) ) ).
% measurable_envelopeD(3)
thf(fact_1078_unsigned__Hahn__decomposition,axiom,
! [N2: sigma_measure_a,M: sigma_measure_a,A: set_a] :
( ( ( sigma_sets_a @ N2 )
= ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ A )
!= top_to1496364449551166952nnreal )
=> ( ( ( sigma_emeasure_a @ N2 @ A )
!= top_to1496364449551166952nnreal )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( sigma_sets_a @ M ) )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ N2 @ 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 @ X2 )
= bot_bot_set_a )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ Xa ) @ ( sigma_emeasure_a @ N2 @ Xa ) ) ) ) ) ) ) ) ) ) ).
% unsigned_Hahn_decomposition
thf(fact_1079_boolean__algebra_Odisj__one__left,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X3 )
= top_top_set_a ) ).
% boolean_algebra.disj_one_left
thf(fact_1080_boolean__algebra_Odisj__one__left,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ top_top_set_set_a @ X3 )
= top_top_set_set_a ) ).
% boolean_algebra.disj_one_left
thf(fact_1081_boolean__algebra_Odisj__one__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ top_top_set_a )
= top_top_set_a ) ).
% boolean_algebra.disj_one_right
thf(fact_1082_boolean__algebra_Odisj__one__right,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ top_top_set_set_a )
= top_top_set_set_a ) ).
% boolean_algebra.disj_one_right
thf(fact_1083_sup__top__left,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X3 )
= top_top_set_a ) ).
% sup_top_left
thf(fact_1084_sup__top__left,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ top_top_set_set_a @ X3 )
= top_top_set_set_a ) ).
% sup_top_left
thf(fact_1085_sup__top__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ top_top_set_a )
= top_top_set_a ) ).
% sup_top_right
thf(fact_1086_sup__top__right,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ top_top_set_set_a )
= top_top_set_set_a ) ).
% sup_top_right
thf(fact_1087_atLeastAtMost__eq__UNIV__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( ( set_or6288561110385358355_set_a @ X3 @ Y2 )
= top_top_set_set_a )
= ( ( X3 = bot_bot_set_a )
& ( Y2 = top_top_set_a ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_1088_measurable__envelopeD_I2_J,axiom,
! [M: sigma_measure_a,A: set_a,E2: set_a] :
( ( comple1317578422686860819lope_a @ M @ A @ E2 )
=> ( member_set_a @ E2 @ ( sigma_sets_a @ M ) ) ) ).
% measurable_envelopeD(2)
thf(fact_1089_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_set_a,X3: set_set_a,Y2: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ X3 )
= bot_bot_set_set_a )
=> ( ( ( sup_sup_set_set_a @ A2 @ X3 )
= top_top_set_set_a )
=> ( ( ( inf_inf_set_set_a @ A2 @ Y2 )
= bot_bot_set_set_a )
=> ( ( ( sup_sup_set_set_a @ A2 @ Y2 )
= top_top_set_set_a )
=> ( X3 = Y2 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1090_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_a,X3: set_a,Y2: set_a] :
( ( ( inf_inf_set_a @ A2 @ X3 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A2 @ X3 )
= top_top_set_a )
=> ( ( ( inf_inf_set_a @ A2 @ Y2 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A2 @ Y2 )
= top_top_set_a )
=> ( X3 = Y2 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1091_measurable__envelopeD1,axiom,
! [M: sigma_measure_a,A: set_a,E2: set_a,F3: set_a] :
( ( comple1317578422686860819lope_a @ M @ A @ E2 )
=> ( ( member_set_a @ F3 @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ F3 @ ( minus_minus_set_a @ E2 @ A ) )
=> ( ( sigma_emeasure_a @ M @ F3 )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% measurable_envelopeD1
thf(fact_1092_measurable__envelope__eq1,axiom,
! [A: set_a,E2: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ A @ E2 )
=> ( ( member_set_a @ E2 @ ( sigma_sets_a @ M ) )
=> ( ( comple1317578422686860819lope_a @ M @ A @ E2 )
= ( ! [X: set_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ X @ ( minus_minus_set_a @ E2 @ A ) )
=> ( ( sigma_emeasure_a @ M @ X )
= zero_z7100319975126383169nnreal ) ) ) ) ) ) ) ).
% measurable_envelope_eq1
thf(fact_1093_finite__unsigned__Hahn__decomposition,axiom,
! [M: sigma_measure_a,N2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M )
=> ( ( measur930452917991658466sure_a @ N2 )
=> ( ( ( sigma_sets_a @ N2 )
= ( sigma_sets_a @ M ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( sigma_sets_a @ M ) )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ N2 @ Xa ) @ ( sigma_emeasure_a @ M @ Xa ) ) ) )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( sigma_sets_a @ M ) )
=> ( ( ( inf_inf_set_a @ Xa @ X2 )
= bot_bot_set_a )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ Xa ) @ ( sigma_emeasure_a @ N2 @ Xa ) ) ) ) ) ) ) ) ).
% finite_unsigned_Hahn_decomposition
thf(fact_1094_UNIV__I,axiom,
! [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_I
thf(fact_1095_UNIV__I,axiom,
! [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_1096_atLeast__empty__triv,axiom,
( ( set_or8362275514725411625_set_a @ bot_bot_set_a )
= top_top_set_set_a ) ).
% atLeast_empty_triv
thf(fact_1097_Diff__UNIV,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_1098_space__borel,axiom,
( ( sigma_space_a @ borel_5459123734250506524orel_a )
= top_top_set_a ) ).
% space_borel
thf(fact_1099_emeasure__empty,axiom,
! [M: sigma_measure_a] :
( ( sigma_emeasure_a @ M @ bot_bot_set_a )
= zero_z7100319975126383169nnreal ) ).
% emeasure_empty
thf(fact_1100_null__setsI,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( ( sigma_emeasure_a @ M @ A )
= zero_z7100319975126383169nnreal )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ A @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_setsI
thf(fact_1101_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_1102_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X: a] : ( member_a @ X @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_1103_top__empty__eq,axiom,
( top_top_set_a_o
= ( ^ [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ) ) ).
% top_empty_eq
thf(fact_1104_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_1105_UNIV__witness,axiom,
? [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_1106_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_1107_UNIV__eq__I,axiom,
! [A: set_set_a] :
( ! [X2: set_a] : ( member_set_a @ X2 @ A )
=> ( top_top_set_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_1108_Un__UNIV__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ top_top_set_a )
= top_top_set_a ) ).
% Un_UNIV_right
thf(fact_1109_Un__UNIV__right,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ top_top_set_set_a )
= top_top_set_set_a ) ).
% Un_UNIV_right
thf(fact_1110_Un__UNIV__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ B )
= top_top_set_a ) ).
% Un_UNIV_left
thf(fact_1111_Un__UNIV__left,axiom,
! [B: set_set_a] :
( ( sup_sup_set_set_a @ top_top_set_set_a @ B )
= top_top_set_set_a ) ).
% Un_UNIV_left
thf(fact_1112_finite__measure_Ofmeasurable__eq__sets,axiom,
! [M: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M )
=> ( ( measur3645360004775918570able_a @ M )
= ( sigma_sets_a @ M ) ) ) ).
% finite_measure.fmeasurable_eq_sets
thf(fact_1113_space__in__borel,axiom,
member_set_a @ top_top_set_a @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ).
% space_in_borel
thf(fact_1114_emeasure__notin__sets,axiom,
! [A: set_a,M: sigma_measure_a] :
( ~ ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ A )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_notin_sets
thf(fact_1115_emeasure__neq__0__sets,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( ( sigma_emeasure_a @ M @ A )
!= zero_z7100319975126383169nnreal )
=> ( member_set_a @ A @ ( sigma_sets_a @ M ) ) ) ).
% emeasure_neq_0_sets
thf(fact_1116_atLeast__eq__UNIV__iff,axiom,
! [X3: set_a] :
( ( ( set_or8362275514725411625_set_a @ X3 )
= top_top_set_set_a )
= ( X3 = bot_bot_set_a ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_1117_emeasure__eq__0,axiom,
! [N2: set_a,M: sigma_measure_a,K2: set_a] :
( ( member_set_a @ N2 @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ N2 )
= zero_z7100319975126383169nnreal )
=> ( ( ord_less_eq_set_a @ K2 @ N2 )
=> ( ( sigma_emeasure_a @ M @ K2 )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% emeasure_eq_0
thf(fact_1118_emeasure__single__in__space,axiom,
! [M: sigma_measure_set_a,X3: set_a] :
( ( ( sigma_emeasure_set_a @ M @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
!= zero_z7100319975126383169nnreal )
=> ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) ) ) ).
% emeasure_single_in_space
thf(fact_1119_emeasure__single__in__space,axiom,
! [M: sigma_measure_a,X3: a] :
( ( ( sigma_emeasure_a @ M @ ( insert_a @ X3 @ bot_bot_set_a ) )
!= zero_z7100319975126383169nnreal )
=> ( member_a @ X3 @ ( sigma_space_a @ M ) ) ) ).
% emeasure_single_in_space
thf(fact_1120_null__part__sets_I2_J,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ ( complete_null_part_a @ M @ S ) )
= zero_z7100319975126383169nnreal ) ) ).
% null_part_sets(2)
thf(fact_1121_null__part__sets_I1_J,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( complete_null_part_a @ M @ S ) @ ( sigma_sets_a @ M ) ) ) ).
% null_part_sets(1)
thf(fact_1122_greaterThanLessThan__empty,axiom,
! [L: a,K: a] :
( ( ord_less_eq_a @ L @ K )
=> ( ( set_or5939364468397584554Than_a @ K @ L )
= bot_bot_set_a ) ) ).
% greaterThanLessThan_empty
thf(fact_1123_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_1124_ivl__disj__int__two_I4_J,axiom,
! [L: a,M2: a,U: a] :
( ( inf_inf_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or5939364468397584554Than_a @ M2 @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(4)
thf(fact_1125_ivl__disj__int__two_I5_J,axiom,
! [L: a,M2: a,U: a] :
( ( inf_inf_set_a @ ( set_or5939364468397584554Than_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(5)
thf(fact_1126_ivl__disj__int__two_I1_J,axiom,
! [L: a,M2: a,U: a] :
( ( inf_inf_set_a @ ( set_or5939364468397584554Than_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(1)
thf(fact_1127_ivl__disj__int__one_I1_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or5939364468397584554Than_a @ L @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(1)
thf(fact_1128_ivl__disj__int__one_I6_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_or5939364468397584554Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(6)
thf(fact_1129_ivl__disj__int__two_I2_J,axiom,
! [L: a,M2: a,U: a] :
( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or5939364468397584554Than_a @ M2 @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(2)
thf(fact_1130_atLeastAtMost__diff__ends,axiom,
! [A2: a,B2: a] :
( ( minus_minus_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( set_or5939364468397584554Than_a @ A2 @ B2 ) ) ).
% atLeastAtMost_diff_ends
thf(fact_1131_emeasure__diff__measure,axiom,
! [M: sigma_measure_a,N2: sigma_measure_a,A: set_a] :
( ( measur930452917991658466sure_a @ M )
=> ( ( measur930452917991658466sure_a @ N2 )
=> ( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N2 ) )
=> ( ! [A7: set_a] :
( ( member_set_a @ A7 @ ( sigma_sets_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ N2 @ A7 ) @ ( sigma_emeasure_a @ M @ A7 ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ ( radon_diff_measure_a @ M @ N2 ) @ A )
= ( minus_8429688780609304081nnreal @ ( sigma_emeasure_a @ M @ A ) @ ( sigma_emeasure_a @ N2 @ A ) ) ) ) ) ) ) ) ).
% emeasure_diff_measure
thf(fact_1132_ivl__disj__un__singleton_I4_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ U ) @ ( insert_a @ U @ bot_bot_set_a ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_1133_ivl__disj__un__two_I5_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_1134_lessThan__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_lessThan_a @ K ) )
= ( ord_less_a @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_1135_lessThan__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_or5421148953861284865_set_a @ K ) )
= ( ord_less_set_a @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_1136_greaterThan__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_or8632414552788122084Than_a @ K ) )
= ( ord_less_a @ K @ I2 ) ) ).
% greaterThan_iff
thf(fact_1137_greaterThan__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_or460448635090783044_set_a @ K ) )
= ( ord_less_set_a @ K @ I2 ) ) ).
% greaterThan_iff
thf(fact_1138_greaterThanLessThan__iff,axiom,
! [I2: a,L: a,U: a] :
( ( member_a @ I2 @ ( set_or5939364468397584554Than_a @ L @ U ) )
= ( ( ord_less_a @ L @ I2 )
& ( ord_less_a @ I2 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_1139_greaterThanLessThan__iff,axiom,
! [I2: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I2 @ ( set_or6017932776736107018_set_a @ L @ U ) )
= ( ( ord_less_set_a @ L @ I2 )
& ( ord_less_set_a @ I2 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_1140_sets__diff__measure,axiom,
! [M: sigma_measure_a,N2: sigma_measure_a] :
( ( sigma_sets_a @ ( radon_diff_measure_a @ M @ N2 ) )
= ( sigma_sets_a @ M ) ) ).
% sets_diff_measure
thf(fact_1141_atLeastatMost__empty,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ( set_or672772299803893939Most_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% atLeastatMost_empty
thf(fact_1142_atLeastLessThan__iff,axiom,
! [I2: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I2 @ ( set_or2348907005316661231_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_set_a @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1143_atLeastLessThan__iff,axiom,
! [I2: a,L: a,U: a] :
( ( member_a @ I2 @ ( set_or5139330845457685135Than_a @ L @ U ) )
= ( ( ord_less_eq_a @ L @ I2 )
& ( ord_less_a @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1144_atLeastLessThan__empty__iff2,axiom,
! [A2: a,B2: a] :
( ( bot_bot_set_a
= ( set_or5139330845457685135Than_a @ A2 @ B2 ) )
= ( ~ ( ord_less_a @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_1145_atLeastLessThan__empty__iff,axiom,
! [A2: a,B2: a] :
( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ~ ( ord_less_a @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_1146_greaterThanAtMost__iff,axiom,
! [I2: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I2 @ ( set_or2503527069484367278_set_a @ L @ U ) )
= ( ( ord_less_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1147_greaterThanAtMost__iff,axiom,
! [I2: a,L: a,U: a] :
( ( member_a @ I2 @ ( set_or4472690218693186638Most_a @ L @ U ) )
= ( ( ord_less_a @ L @ I2 )
& ( ord_less_eq_a @ I2 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1148_greaterThanAtMost__empty__iff2,axiom,
! [K: a,L: a] :
( ( bot_bot_set_a
= ( set_or4472690218693186638Most_a @ K @ L ) )
= ( ~ ( ord_less_a @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_1149_greaterThanAtMost__empty__iff,axiom,
! [K: a,L: a] :
( ( ( set_or4472690218693186638Most_a @ K @ L )
= bot_bot_set_a )
= ( ~ ( ord_less_a @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_1150_order__le__imp__less__or__eq,axiom,
! [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( ord_less_a @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1151_linorder__le__less__linear,axiom,
! [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
| ( ord_less_a @ Y2 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_1152_order__less__le__subst1,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1153_order__le__less__subst2,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1154_order__less__le__trans,axiom,
! [X3: a,Y2: a,Z: a] :
( ( ord_less_a @ X3 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z )
=> ( ord_less_a @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1155_order__le__less__trans,axiom,
! [X3: a,Y2: a,Z: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( ord_less_a @ Y2 @ Z )
=> ( ord_less_a @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1156_order__neq__le__trans,axiom,
! [A2: a,B2: a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_a @ A2 @ B2 )
=> ( ord_less_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1157_order__le__neq__trans,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1158_order__less__imp__le,axiom,
! [X3: a,Y2: a] :
( ( ord_less_a @ X3 @ Y2 )
=> ( ord_less_eq_a @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1159_linorder__not__less,axiom,
! [X3: a,Y2: a] :
( ( ~ ( ord_less_a @ X3 @ Y2 ) )
= ( ord_less_eq_a @ Y2 @ X3 ) ) ).
% linorder_not_less
thf(fact_1160_linorder__not__le,axiom,
! [X3: a,Y2: a] :
( ( ~ ( ord_less_eq_a @ X3 @ Y2 ) )
= ( ord_less_a @ Y2 @ X3 ) ) ).
% linorder_not_le
thf(fact_1161_order__less__le,axiom,
( ord_less_a
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_1162_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_1163_dual__order_Ostrict__implies__order,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ord_less_eq_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1164_order_Ostrict__implies__order,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ord_less_eq_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1165_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B5: a,A5: a] :
( ( ord_less_eq_a @ B5 @ A5 )
& ~ ( ord_less_eq_a @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1166_dual__order_Ostrict__trans2,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ C @ B2 )
=> ( ord_less_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1167_dual__order_Ostrict__trans1,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_a @ C @ B2 )
=> ( ord_less_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1168_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B5: a,A5: a] :
( ( ord_less_eq_a @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1169_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B5: a,A5: a] :
( ( ord_less_a @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1170_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A5: a,B5: a] :
( ( ord_less_eq_a @ A5 @ B5 )
& ~ ( ord_less_eq_a @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1171_order_Ostrict__trans2,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1172_order_Ostrict__trans1,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1173_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A5: a,B5: a] :
( ( ord_less_eq_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1174_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A5: a,B5: a] :
( ( ord_less_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1175_not__le__imp__less,axiom,
! [Y2: a,X3: a] :
( ~ ( ord_less_eq_a @ Y2 @ X3 )
=> ( ord_less_a @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_1176_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ~ ( ord_less_eq_a @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_1177_antisym__conv2,axiom,
! [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( ~ ( ord_less_a @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1178_antisym__conv1,axiom,
! [X3: a,Y2: a] :
( ~ ( ord_less_a @ X3 @ Y2 )
=> ( ( ord_less_eq_a @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1179_nless__le,axiom,
! [A2: a,B2: a] :
( ( ~ ( ord_less_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1180_leI,axiom,
! [X3: a,Y2: a] :
( ~ ( ord_less_a @ X3 @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X3 ) ) ).
% leI
thf(fact_1181_leD,axiom,
! [Y2: a,X3: a] :
( ( ord_less_eq_a @ Y2 @ X3 )
=> ~ ( ord_less_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_1182_atLeastatMost__psubset__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ord_less_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or672772299803893939Most_a @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D2 )
& ( ( ord_less_a @ C @ A2 )
| ( ord_less_a @ B2 @ D2 ) ) ) )
& ( ord_less_eq_a @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1183_bot_Onot__eq__extremum,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1184_bot_Oextremum__strict,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_1185_not__psubset__empty,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_1186_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1187_sup_Ostrict__coboundedI2,axiom,
! [C: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1188_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ C @ A2 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1189_sup_Ostrict__coboundedI1,axiom,
! [C: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ C @ A2 )
=> ( ord_less_set_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1190_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B5: set_a,A5: set_a] :
( ( A5
= ( sup_sup_set_a @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1191_sup_Ostrict__order__iff,axiom,
( ord_less_set_set_a
= ( ^ [B5: set_set_a,A5: set_set_a] :
( ( A5
= ( sup_sup_set_set_a @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1192_sup_Ostrict__boundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_1193_sup_Ostrict__boundedE,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_set_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_1194_sup_Oabsorb4,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_1195_sup_Oabsorb4,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_1196_sup_Oabsorb3,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_1197_sup_Oabsorb3,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_1198_less__supI2,axiom,
! [X3: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ X3 @ B2 )
=> ( ord_less_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1199_less__supI2,axiom,
! [X3: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ X3 @ B2 )
=> ( ord_less_set_set_a @ X3 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1200_less__supI1,axiom,
! [X3: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ X3 @ A2 )
=> ( ord_less_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1201_less__supI1,axiom,
! [X3: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ X3 @ A2 )
=> ( ord_less_set_set_a @ X3 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1202_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B6: a] : ( member_a @ B6 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1203_psubset__imp__ex__mem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ? [B6: set_a] : ( member_set_a @ B6 @ ( minus_5736297505244876581_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1204_less__eq__measure_Osimps,axiom,
( ord_le254669795585780187sure_a
= ( ^ [A1: sigma_measure_a,A22: sigma_measure_a] :
( ? [M4: sigma_measure_a,N3: sigma_measure_a] :
( ( A1 = M4 )
& ( A22 = N3 )
& ( ord_less_set_a @ ( sigma_space_a @ M4 ) @ ( sigma_space_a @ N3 ) ) )
| ? [M4: sigma_measure_a,N3: sigma_measure_a] :
( ( A1 = M4 )
& ( A22 = N3 )
& ( ( sigma_space_a @ M4 )
= ( sigma_space_a @ N3 ) )
& ( ord_less_set_set_a @ ( sigma_sets_a @ M4 ) @ ( sigma_sets_a @ N3 ) ) )
| ? [M4: sigma_measure_a,N3: sigma_measure_a] :
( ( A1 = M4 )
& ( A22 = N3 )
& ( ( sigma_space_a @ M4 )
= ( sigma_space_a @ N3 ) )
& ( ( sigma_sets_a @ M4 )
= ( sigma_sets_a @ N3 ) )
& ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ M4 ) @ ( sigma_emeasure_a @ N3 ) ) ) ) ) ) ).
% less_eq_measure.simps
thf(fact_1205_less__eq__measure_Ocases,axiom,
! [A12: sigma_measure_a,A23: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A12 @ A23 )
=> ( ~ ( ord_less_set_a @ ( sigma_space_a @ A12 ) @ ( sigma_space_a @ A23 ) )
=> ( ( ( ( sigma_space_a @ A12 )
= ( sigma_space_a @ A23 ) )
=> ~ ( ord_less_set_set_a @ ( sigma_sets_a @ A12 ) @ ( sigma_sets_a @ A23 ) ) )
=> ~ ( ( ( sigma_space_a @ A12 )
= ( sigma_space_a @ A23 ) )
=> ( ( ( sigma_sets_a @ A12 )
= ( sigma_sets_a @ A23 ) )
=> ~ ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ A12 ) @ ( sigma_emeasure_a @ A23 ) ) ) ) ) ) ) ).
% less_eq_measure.cases
thf(fact_1206_Iic__subset__Iio__iff,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ A2 ) @ ( set_ord_lessThan_a @ B2 ) )
= ( ord_less_a @ A2 @ B2 ) ) ).
% Iic_subset_Iio_iff
thf(fact_1207_Ici__subset__Ioi__iff,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ A2 ) @ ( set_or8632414552788122084Than_a @ B2 ) )
= ( ord_less_a @ B2 @ A2 ) ) ).
% Ici_subset_Ioi_iff
thf(fact_1208_less__eq__measure_Ointros_I2_J,axiom,
! [M: sigma_measure_a,N2: sigma_measure_a] :
( ( ( sigma_space_a @ M )
= ( sigma_space_a @ N2 ) )
=> ( ( ord_less_set_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ N2 ) )
=> ( ord_le254669795585780187sure_a @ M @ N2 ) ) ) ).
% less_eq_measure.intros(2)
thf(fact_1209_psubset__insert__iff,axiom,
! [A: set_set_a,X3: set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ ( insert_set_a @ X3 @ B ) )
= ( ( ( member_set_a @ X3 @ B )
=> ( ord_less_set_set_a @ A @ B ) )
& ( ~ ( member_set_a @ X3 @ B )
=> ( ( ( member_set_a @ X3 @ A )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X3 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1210_psubset__insert__iff,axiom,
! [A: set_a,X3: a,B: set_a] :
( ( ord_less_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( ( ( member_a @ X3 @ B )
=> ( ord_less_set_a @ A @ B ) )
& ( ~ ( member_a @ X3 @ B )
=> ( ( ( member_a @ X3 @ A )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1211_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or5139330845457685135Than_a @ C @ D2 ) )
= ( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_a @ B2 @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1212_Iio__Int__singleton,axiom,
! [X3: a,K: a] :
( ( ( ord_less_a @ X3 @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
= ( insert_a @ X3 @ bot_bot_set_a ) ) )
& ( ~ ( ord_less_a @ X3 @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
= bot_bot_set_a ) ) ) ).
% Iio_Int_singleton
thf(fact_1213_ivl__disj__un__two__touch_I2_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_1214_less__separate,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ? [A6: set_a,B6: set_a] :
( ( member_set_a @ X3 @ ( set_or5421148953861284865_set_a @ A6 ) )
& ( member_set_a @ Y2 @ ( set_or460448635090783044_set_a @ B6 ) )
& ( ( inf_inf_set_set_a @ ( set_or5421148953861284865_set_a @ A6 ) @ ( set_or460448635090783044_set_a @ B6 ) )
= bot_bot_set_set_a ) ) ) ).
% less_separate
thf(fact_1215_less__separate,axiom,
! [X3: a,Y2: a] :
( ( ord_less_a @ X3 @ Y2 )
=> ? [A6: a,B6: a] :
( ( member_a @ X3 @ ( set_ord_lessThan_a @ A6 ) )
& ( member_a @ Y2 @ ( set_or8632414552788122084Than_a @ B6 ) )
& ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ A6 ) @ ( set_or8632414552788122084Than_a @ B6 ) )
= bot_bot_set_a ) ) ) ).
% less_separate
thf(fact_1216_ivl__disj__un__two__touch_I3_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_1217_ivl__disj__un__two_I1_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_1218_ivl__disj__un__two_I2_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or5939364468397584554Than_a @ M2 @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_1219_ivl__disj__un__one_I1_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or5939364468397584554Than_a @ L @ U ) )
= ( set_ord_lessThan_a @ U ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_1220_ivl__disj__un__two__touch_I1_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_1221_ivl__disj__un__one_I6_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= ( set_or8632414552788122084Than_a @ L ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_1222_ivl__disj__un__two_I4_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or5939364468397584554Than_a @ M2 @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_1223_ivl__disj__un__singleton_I3_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( insert_a @ L @ bot_bot_set_a ) @ ( set_or5939364468397584554Than_a @ L @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_1224_minf_I8_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ X5 @ Z3 )
=> ~ ( ord_less_eq_a @ T3 @ X5 ) ) ).
% minf(8)
thf(fact_1225_minf_I6_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ X5 @ Z3 )
=> ( ord_less_eq_a @ X5 @ T3 ) ) ).
% minf(6)
thf(fact_1226_pinf_I8_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ Z3 @ X5 )
=> ( ord_less_eq_a @ T3 @ X5 ) ) ).
% pinf(8)
thf(fact_1227_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_1228_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_1229_pinf_I6_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X5: a] :
( ( ord_less_a @ Z3 @ X5 )
=> ~ ( ord_less_eq_a @ X5 @ T3 ) ) ).
% pinf(6)
thf(fact_1230_verit__comp__simplify1_I3_J,axiom,
! [B8: a,A8: a] :
( ( ~ ( ord_less_eq_a @ B8 @ A8 ) )
= ( ord_less_a @ A8 @ B8 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1231_verit__comp__simplify1_I2_J,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1232_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_1233_measurable__envelope__eq2,axiom,
! [A: set_a,E2: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ A @ E2 )
=> ( ( member_set_a @ E2 @ ( sigma_sets_a @ M ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ E2 ) @ extend2057119558705770725nnreal )
=> ( ( comple1317578422686860819lope_a @ M @ A @ E2 )
= ( ( sigma_emeasure_a @ M @ E2 )
= ( comple3326625653960060650e_of_a @ M @ A ) ) ) ) ) ) ).
% measurable_envelope_eq2
thf(fact_1234_cld__measure_Onotin__sets__outer__measure__of__cover,axiom,
! [M: sigma_measure_a,E2: set_a] :
( ( comple4590207612277428097sure_a @ M )
=> ( ( ord_less_eq_set_a @ E2 @ ( sigma_space_a @ M ) )
=> ( ~ ( member_set_a @ E2 @ ( sigma_sets_a @ M ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( sigma_sets_a @ M ) )
& ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( sigma_emeasure_a @ M @ X2 ) )
& ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ X2 ) @ extend2057119558705770725nnreal )
& ( ( comple3326625653960060650e_of_a @ M @ ( inf_inf_set_a @ X2 @ E2 ) )
= ( sigma_emeasure_a @ M @ X2 ) )
& ( ( comple3326625653960060650e_of_a @ M @ ( minus_minus_set_a @ X2 @ E2 ) )
= ( sigma_emeasure_a @ M @ X2 ) ) ) ) ) ) ).
% cld_measure.notin_sets_outer_measure_of_cover
thf(fact_1235_fmeasurableI,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ A ) @ extend2057119558705770725nnreal )
=> ( member_set_a @ A @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurableI
thf(fact_1236_emeasure__Diff,axiom,
! [M: sigma_measure_a,B: set_a,A: set_a] :
( ( ( sigma_emeasure_a @ M @ B )
!= extend2057119558705770725nnreal )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( sigma_emeasure_a @ M @ ( minus_minus_set_a @ A @ B ) )
= ( minus_8429688780609304081nnreal @ ( sigma_emeasure_a @ M @ A ) @ ( sigma_emeasure_a @ M @ B ) ) ) ) ) ) ) ).
% emeasure_Diff
thf(fact_1237_emeasure__compl,axiom,
! [S2: set_a,M: sigma_measure_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ S2 )
!= extend2057119558705770725nnreal )
=> ( ( sigma_emeasure_a @ M @ ( minus_minus_set_a @ ( sigma_space_a @ M ) @ S2 ) )
= ( minus_8429688780609304081nnreal @ ( sigma_emeasure_a @ M @ ( sigma_space_a @ M ) ) @ ( sigma_emeasure_a @ M @ S2 ) ) ) ) ) ).
% emeasure_compl
thf(fact_1238_locally__determined__measure__axioms_Ointro,axiom,
! [M: sigma_measure_a] :
( ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ ( sigma_space_a @ M ) )
=> ( ! [B9: set_a] :
( ( member_set_a @ B9 @ ( sigma_sets_a @ M ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ B9 ) @ extend2057119558705770725nnreal )
=> ( member_set_a @ ( inf_inf_set_a @ A7 @ B9 ) @ ( sigma_sets_a @ M ) ) ) )
=> ( member_set_a @ A7 @ ( sigma_sets_a @ M ) ) ) )
=> ( comple7282333422040910444ioms_a @ M ) ) ).
% locally_determined_measure_axioms.intro
thf(fact_1239_locally__determined__measure__axioms__def,axiom,
( comple7282333422040910444ioms_a
= ( ^ [M4: sigma_measure_a] :
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( sigma_space_a @ M4 ) )
=> ( ! [B3: set_a] :
( ( member_set_a @ B3 @ ( sigma_sets_a @ M4 ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M4 @ B3 ) @ extend2057119558705770725nnreal )
=> ( member_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( sigma_sets_a @ M4 ) ) ) )
=> ( member_set_a @ A3 @ ( sigma_sets_a @ M4 ) ) ) ) ) ) ).
% locally_determined_measure_axioms_def
thf(fact_1240_locally__determined__measure_Olocally__determined,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( comple7257097362500340559sure_a @ M )
=> ( ( ord_less_eq_set_a @ A @ ( sigma_space_a @ M ) )
=> ( ! [B7: set_a] :
( ( member_set_a @ B7 @ ( sigma_sets_a @ M ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ B7 ) @ extend2057119558705770725nnreal )
=> ( member_set_a @ ( inf_inf_set_a @ A @ B7 ) @ ( sigma_sets_a @ M ) ) ) )
=> ( member_set_a @ A @ ( sigma_sets_a @ M ) ) ) ) ) ).
% locally_determined_measure.locally_determined
thf(fact_1241_semifinite__measure__def,axiom,
( comple8962655729304856821sure_a
= ( ^ [M4: sigma_measure_a] :
! [A3: set_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ M4 ) )
=> ( ( ( sigma_emeasure_a @ M4 @ A3 )
= extend2057119558705770725nnreal )
=> ? [X: set_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M4 ) )
& ( ord_less_eq_set_a @ X @ A3 )
& ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M4 @ X ) @ extend2057119558705770725nnreal ) ) ) ) ) ) ).
% semifinite_measure_def
thf(fact_1242_semifinite__measure_Ointro,axiom,
! [M: sigma_measure_a] :
( ! [A7: set_a] :
( ( member_set_a @ A7 @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ A7 )
= extend2057119558705770725nnreal )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ ( sigma_sets_a @ M ) )
& ( ord_less_eq_set_a @ X5 @ A7 )
& ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ X5 ) @ extend2057119558705770725nnreal ) ) ) )
=> ( comple8962655729304856821sure_a @ M ) ) ).
% semifinite_measure.intro
thf(fact_1243_semifinite__measure_Osemifinite,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( comple8962655729304856821sure_a @ M )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ A )
= extend2057119558705770725nnreal )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( sigma_sets_a @ M ) )
& ( ord_less_eq_set_a @ X2 @ A )
& ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ X2 ) @ extend2057119558705770725nnreal ) ) ) ) ) ).
% semifinite_measure.semifinite
thf(fact_1244_complete__measure_Ocomplete__sets__sandwich,axiom,
! [M: sigma_measure_a,A: set_a,C2: set_a,B: set_a] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ C2 @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( ( sigma_emeasure_a @ M @ A )
= ( sigma_emeasure_a @ M @ C2 ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ A ) @ extend2057119558705770725nnreal )
=> ( member_set_a @ B @ ( sigma_sets_a @ M ) ) ) ) ) ) ) ) ) ).
% complete_measure.complete_sets_sandwich
thf(fact_1245_emeasure__Un_H,axiom,
! [A: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ ( inf_inf_set_a @ A @ B ) @ ( measure_null_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ ( sup_sup_set_a @ A @ B ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_a @ M @ A ) @ ( sigma_emeasure_a @ M @ B ) ) ) ) ) ) ).
% emeasure_Un'
thf(fact_1246_emeasure__Un_H,axiom,
! [A: set_set_a,M: sigma_measure_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a @ B @ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( measur1516554128032400785_set_a @ M ) )
=> ( ( sigma_emeasure_set_a @ M @ ( sup_sup_set_set_a @ A @ B ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_set_a @ M @ A ) @ ( sigma_emeasure_set_a @ M @ B ) ) ) ) ) ) ).
% emeasure_Un'
thf(fact_1247_complete__measure_Ocomplete2,axiom,
! [M: sigma_measure_a,A: set_a,B: set_a] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ A @ ( measure_null_sets_a @ M ) ) ) ) ) ).
% complete_measure.complete2
thf(fact_1248_emeasure__insert,axiom,
! [X3: set_a,M: sigma_measure_set_a,A: set_set_a] :
( ( member_set_set_a @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) @ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a @ A @ ( sigma_sets_set_a @ M ) )
=> ( ~ ( member_set_a @ X3 @ A )
=> ( ( sigma_emeasure_set_a @ M @ ( insert_set_a @ X3 @ A ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_set_a @ M @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) @ ( sigma_emeasure_set_a @ M @ A ) ) ) ) ) ) ).
% emeasure_insert
thf(fact_1249_emeasure__insert,axiom,
! [X3: a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ~ ( member_a @ X3 @ A )
=> ( ( sigma_emeasure_a @ M @ ( insert_a @ X3 @ A ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_a @ M @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ ( sigma_emeasure_a @ M @ A ) ) ) ) ) ) ).
% emeasure_insert
thf(fact_1250_emeasure__insert__ne,axiom,
! [A: set_set_a,X3: set_a,M: sigma_measure_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ( member_set_set_a @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) @ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a @ A @ ( sigma_sets_set_a @ M ) )
=> ( ~ ( member_set_a @ X3 @ A )
=> ( ( sigma_emeasure_set_a @ M @ ( insert_set_a @ X3 @ A ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_set_a @ M @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) @ ( sigma_emeasure_set_a @ M @ A ) ) ) ) ) ) ) ).
% emeasure_insert_ne
thf(fact_1251_emeasure__insert__ne,axiom,
! [A: set_a,X3: a,M: sigma_measure_a] :
( ( A != bot_bot_set_a )
=> ( ( member_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ~ ( member_a @ X3 @ A )
=> ( ( sigma_emeasure_a @ M @ ( insert_a @ X3 @ A ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_a @ M @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ ( sigma_emeasure_a @ M @ A ) ) ) ) ) ) ) ).
% emeasure_insert_ne
thf(fact_1252_emeasure__Un__Int,axiom,
! [A: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_a @ M @ A ) @ ( sigma_emeasure_a @ M @ B ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_a @ M @ ( sup_sup_set_a @ A @ B ) ) @ ( sigma_emeasure_a @ M @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ) ).
% emeasure_Un_Int
thf(fact_1253_emeasure__Un__Int,axiom,
! [A: set_set_a,M: sigma_measure_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a @ B @ ( sigma_sets_set_a @ M ) )
=> ( ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_set_a @ M @ A ) @ ( sigma_emeasure_set_a @ M @ B ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_set_a @ M @ ( sup_sup_set_set_a @ A @ B ) ) @ ( sigma_emeasure_set_a @ M @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ) ) ).
% emeasure_Un_Int
thf(fact_1254_complete__measure__def,axiom,
( comple8155536527497655953sure_a
= ( ^ [M4: sigma_measure_a] :
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( member_set_a @ A3 @ ( measure_null_sets_a @ M4 ) )
=> ( member_set_a @ B3 @ ( sigma_sets_a @ M4 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_1255_complete__measure_Ocomplete,axiom,
! [M: sigma_measure_a,B: set_a,A: set_a] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ B @ ( sigma_sets_a @ M ) ) ) ) ) ).
% complete_measure.complete
thf(fact_1256_complete__measure_Ointro,axiom,
! [M: sigma_measure_a] :
( ! [A7: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ A7 )
=> ( ( member_set_a @ A7 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ B7 @ ( sigma_sets_a @ M ) ) ) )
=> ( comple8155536527497655953sure_a @ M ) ) ).
% complete_measure.intro
thf(fact_1257_emeasure__Un,axiom,
! [A: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ ( sup_sup_set_a @ A @ B ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_a @ M @ A ) @ ( sigma_emeasure_a @ M @ ( minus_minus_set_a @ B @ A ) ) ) ) ) ) ).
% emeasure_Un
thf(fact_1258_emeasure__Un,axiom,
! [A: set_set_a,M: sigma_measure_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a @ B @ ( sigma_sets_set_a @ M ) )
=> ( ( sigma_emeasure_set_a @ M @ ( sup_sup_set_set_a @ A @ B ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_set_a @ M @ A ) @ ( sigma_emeasure_set_a @ M @ ( minus_5736297505244876581_set_a @ B @ A ) ) ) ) ) ) ).
% emeasure_Un
thf(fact_1259_emeasure__subadditive,axiom,
! [A: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ ( sup_sup_set_a @ A @ B ) ) @ ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_a @ M @ A ) @ ( sigma_emeasure_a @ M @ B ) ) ) ) ) ).
% emeasure_subadditive
thf(fact_1260_emeasure__subadditive,axiom,
! [A: set_set_a,M: sigma_measure_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a @ B @ ( sigma_sets_set_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_set_a @ M @ ( sup_sup_set_set_a @ A @ B ) ) @ ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_set_a @ M @ A ) @ ( sigma_emeasure_set_a @ M @ B ) ) ) ) ) ).
% emeasure_subadditive
thf(fact_1261_plus__emeasure,axiom,
! [A2: set_set_a,M: sigma_measure_set_a,B2: set_set_a] :
( ( member_set_set_a @ A2 @ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a @ B2 @ ( sigma_sets_set_a @ M ) )
=> ( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
=> ( ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_set_a @ M @ A2 ) @ ( sigma_emeasure_set_a @ M @ B2 ) )
= ( sigma_emeasure_set_a @ M @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% plus_emeasure
thf(fact_1262_plus__emeasure,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 ) )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_a @ M @ A2 ) @ ( sigma_emeasure_a @ M @ B2 ) )
= ( sigma_emeasure_a @ M @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% plus_emeasure
thf(fact_1263_emeasure__additive,axiom,
! [M: sigma_measure_a] : ( measur2545585918067192918nnreal @ ( sigma_sets_a @ M ) @ ( sigma_emeasure_a @ M ) ) ).
% emeasure_additive
thf(fact_1264_null__sets_Ovolume__additiveI,axiom,
! [M: sigma_measure_set_a,Mu: set_set_a > extend8495563244428889912nnreal] :
( ! [A6: set_set_a] :
( ( member_set_set_a @ A6 @ ( measur1516554128032400785_set_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( Mu @ A6 ) ) )
=> ( ( ( Mu @ bot_bot_set_set_a )
= zero_z7100319975126383169nnreal )
=> ( ! [A6: set_set_a,B6: set_set_a] :
( ( member_set_set_a @ A6 @ ( measur1516554128032400785_set_a @ M ) )
=> ( ( member_set_set_a @ B6 @ ( measur1516554128032400785_set_a @ M ) )
=> ( ( ( inf_inf_set_set_a @ A6 @ B6 )
= bot_bot_set_set_a )
=> ( ( Mu @ ( sup_sup_set_set_a @ A6 @ B6 ) )
= ( plus_p1859984266308609217nnreal @ ( Mu @ A6 ) @ ( Mu @ B6 ) ) ) ) ) )
=> ( volume_set_a @ ( measur1516554128032400785_set_a @ M ) @ Mu ) ) ) ) ).
% null_sets.volume_additiveI
thf(fact_1265_null__sets_Ovolume__additiveI,axiom,
! [M: sigma_measure_a,Mu: set_a > extend8495563244428889912nnreal] :
( ! [A6: set_a] :
( ( member_set_a @ A6 @ ( measure_null_sets_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( Mu @ A6 ) ) )
=> ( ( ( Mu @ bot_bot_set_a )
= zero_z7100319975126383169nnreal )
=> ( ! [A6: set_a,B6: set_a] :
( ( member_set_a @ A6 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ B6 @ ( measure_null_sets_a @ M ) )
=> ( ( ( inf_inf_set_a @ A6 @ B6 )
= bot_bot_set_a )
=> ( ( Mu @ ( sup_sup_set_a @ A6 @ B6 ) )
= ( plus_p1859984266308609217nnreal @ ( Mu @ A6 ) @ ( Mu @ B6 ) ) ) ) ) )
=> ( volume_a @ ( measure_null_sets_a @ M ) @ Mu ) ) ) ) ).
% null_sets.volume_additiveI
thf(fact_1266_volume__empty,axiom,
! [M: set_set_a,F: set_a > extend8495563244428889912nnreal] :
( ( volume_a @ M @ F )
=> ( ( F @ bot_bot_set_a )
= zero_z7100319975126383169nnreal ) ) ).
% volume_empty
thf(fact_1267_sets_Oextend__volume,axiom,
! [M: sigma_measure_a,Mu: set_a > extend8495563244428889912nnreal] :
( ( volume_a @ ( sigma_sets_a @ M ) @ Mu )
=> ? [Mu2: set_a > extend8495563244428889912nnreal] :
( ( volume_a @ ( sigma_6990475660951004707ring_a @ ( sigma_sets_a @ M ) ) @ Mu2 )
& ! [X5: set_a] :
( ( member_set_a @ X5 @ ( sigma_sets_a @ M ) )
=> ( ( Mu2 @ X5 )
= ( Mu @ X5 ) ) ) ) ) ).
% sets.extend_volume
thf(fact_1268_sets_Ovolume__additiveI,axiom,
! [M: sigma_measure_set_a,Mu: set_set_a > extend8495563244428889912nnreal] :
( ! [A6: set_set_a] :
( ( member_set_set_a @ A6 @ ( sigma_sets_set_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( Mu @ A6 ) ) )
=> ( ( ( Mu @ bot_bot_set_set_a )
= zero_z7100319975126383169nnreal )
=> ( ! [A6: set_set_a,B6: set_set_a] :
( ( member_set_set_a @ A6 @ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a @ B6 @ ( sigma_sets_set_a @ M ) )
=> ( ( ( inf_inf_set_set_a @ A6 @ B6 )
= bot_bot_set_set_a )
=> ( ( Mu @ ( sup_sup_set_set_a @ A6 @ B6 ) )
= ( plus_p1859984266308609217nnreal @ ( Mu @ A6 ) @ ( Mu @ B6 ) ) ) ) ) )
=> ( volume_set_a @ ( sigma_sets_set_a @ M ) @ Mu ) ) ) ) ).
% sets.volume_additiveI
thf(fact_1269_sets_Ovolume__additiveI,axiom,
! [M: sigma_measure_a,Mu: set_a > extend8495563244428889912nnreal] :
( ! [A6: set_a] :
( ( member_set_a @ A6 @ ( sigma_sets_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( Mu @ A6 ) ) )
=> ( ( ( Mu @ bot_bot_set_a )
= zero_z7100319975126383169nnreal )
=> ( ! [A6: set_a,B6: set_a] :
( ( member_set_a @ A6 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B6 @ ( sigma_sets_a @ M ) )
=> ( ( ( inf_inf_set_a @ A6 @ B6 )
= bot_bot_set_a )
=> ( ( Mu @ ( sup_sup_set_a @ A6 @ B6 ) )
= ( plus_p1859984266308609217nnreal @ ( Mu @ A6 ) @ ( Mu @ B6 ) ) ) ) ) )
=> ( volume_a @ ( sigma_sets_a @ M ) @ Mu ) ) ) ) ).
% sets.volume_additiveI
thf(fact_1270_fmeasurable_Ovolume__additiveI,axiom,
! [M: sigma_measure_set_a,Mu: set_set_a > extend8495563244428889912nnreal] :
( ! [A6: set_set_a] :
( ( member_set_set_a @ A6 @ ( measur7460903245211743562_set_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( Mu @ A6 ) ) )
=> ( ( ( Mu @ bot_bot_set_set_a )
= zero_z7100319975126383169nnreal )
=> ( ! [A6: set_set_a,B6: set_set_a] :
( ( member_set_set_a @ A6 @ ( measur7460903245211743562_set_a @ M ) )
=> ( ( member_set_set_a @ B6 @ ( measur7460903245211743562_set_a @ M ) )
=> ( ( ( inf_inf_set_set_a @ A6 @ B6 )
= bot_bot_set_set_a )
=> ( ( Mu @ ( sup_sup_set_set_a @ A6 @ B6 ) )
= ( plus_p1859984266308609217nnreal @ ( Mu @ A6 ) @ ( Mu @ B6 ) ) ) ) ) )
=> ( volume_set_a @ ( measur7460903245211743562_set_a @ M ) @ Mu ) ) ) ) ).
% fmeasurable.volume_additiveI
thf(fact_1271_fmeasurable_Ovolume__additiveI,axiom,
! [M: sigma_measure_a,Mu: set_a > extend8495563244428889912nnreal] :
( ! [A6: set_a] :
( ( member_set_a @ A6 @ ( measur3645360004775918570able_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( Mu @ A6 ) ) )
=> ( ( ( Mu @ bot_bot_set_a )
= zero_z7100319975126383169nnreal )
=> ( ! [A6: set_a,B6: set_a] :
( ( member_set_a @ A6 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ B6 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( ( inf_inf_set_a @ A6 @ B6 )
= bot_bot_set_a )
=> ( ( Mu @ ( sup_sup_set_a @ A6 @ B6 ) )
= ( plus_p1859984266308609217nnreal @ ( Mu @ A6 ) @ ( Mu @ B6 ) ) ) ) ) )
=> ( volume_a @ ( measur3645360004775918570able_a @ M ) @ Mu ) ) ) ) ).
% fmeasurable.volume_additiveI
thf(fact_1272_ring__of__sets_Ovolume__additiveI,axiom,
! [Omega: set_set_a,M: set_set_set_a,Mu: set_set_a > extend8495563244428889912nnreal] :
( ( sigma_5899762507328327638_set_a @ Omega @ M )
=> ( ! [A6: set_set_a] :
( ( member_set_set_a @ A6 @ M )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( Mu @ A6 ) ) )
=> ( ( ( Mu @ bot_bot_set_set_a )
= zero_z7100319975126383169nnreal )
=> ( ! [A6: set_set_a,B6: set_set_a] :
( ( member_set_set_a @ A6 @ M )
=> ( ( member_set_set_a @ B6 @ M )
=> ( ( ( inf_inf_set_set_a @ A6 @ B6 )
= bot_bot_set_set_a )
=> ( ( Mu @ ( sup_sup_set_set_a @ A6 @ B6 ) )
= ( plus_p1859984266308609217nnreal @ ( Mu @ A6 ) @ ( Mu @ B6 ) ) ) ) ) )
=> ( volume_set_a @ M @ Mu ) ) ) ) ) ).
% ring_of_sets.volume_additiveI
thf(fact_1273_ring__of__sets_Ovolume__additiveI,axiom,
! [Omega: set_a,M: set_set_a,Mu: set_a > extend8495563244428889912nnreal] :
( ( sigma_ring_of_sets_a @ Omega @ M )
=> ( ! [A6: set_a] :
( ( member_set_a @ A6 @ M )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( Mu @ A6 ) ) )
=> ( ( ( Mu @ bot_bot_set_a )
= zero_z7100319975126383169nnreal )
=> ( ! [A6: set_a,B6: set_a] :
( ( member_set_a @ A6 @ M )
=> ( ( member_set_a @ B6 @ M )
=> ( ( ( inf_inf_set_a @ A6 @ B6 )
= bot_bot_set_a )
=> ( ( Mu @ ( sup_sup_set_a @ A6 @ B6 ) )
= ( plus_p1859984266308609217nnreal @ ( Mu @ A6 ) @ ( Mu @ B6 ) ) ) ) ) )
=> ( volume_a @ M @ Mu ) ) ) ) ) ).
% ring_of_sets.volume_additiveI
thf(fact_1274_sets_Olambda__system__strong__additive,axiom,
! [Z: set_set_a,M: sigma_measure_set_a,X3: set_set_a,Y2: set_set_a,F: set_set_a > extend8495563244428889912nnreal] :
( ( member_set_set_a @ Z @ ( sigma_sets_set_a @ M ) )
=> ( ( ( inf_inf_set_set_a @ X3 @ Y2 )
= bot_bot_set_set_a )
=> ( ( member_set_set_a @ X3 @ ( lambda_system_set_a @ ( sigma_space_set_a @ M ) @ ( sigma_sets_set_a @ M ) @ F ) )
=> ( ( member_set_set_a @ Y2 @ ( lambda_system_set_a @ ( sigma_space_set_a @ M ) @ ( sigma_sets_set_a @ M ) @ F ) )
=> ( ( F @ ( inf_inf_set_set_a @ Z @ ( sup_sup_set_set_a @ X3 @ Y2 ) ) )
= ( plus_p1859984266308609217nnreal @ ( F @ ( inf_inf_set_set_a @ Z @ X3 ) ) @ ( F @ ( inf_inf_set_set_a @ Z @ Y2 ) ) ) ) ) ) ) ) ).
% sets.lambda_system_strong_additive
thf(fact_1275_sets_Olambda__system__strong__additive,axiom,
! [Z: set_a,M: sigma_measure_a,X3: set_a,Y2: set_a,F: set_a > extend8495563244428889912nnreal] :
( ( member_set_a @ Z @ ( sigma_sets_a @ M ) )
=> ( ( ( inf_inf_set_a @ X3 @ Y2 )
= bot_bot_set_a )
=> ( ( member_set_a @ X3 @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) )
=> ( ( member_set_a @ Y2 @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) )
=> ( ( F @ ( inf_inf_set_a @ Z @ ( sup_sup_set_a @ X3 @ Y2 ) ) )
= ( plus_p1859984266308609217nnreal @ ( F @ ( inf_inf_set_a @ Z @ X3 ) ) @ ( F @ ( inf_inf_set_a @ Z @ Y2 ) ) ) ) ) ) ) ) ).
% sets.lambda_system_strong_additive
thf(fact_1276_sets_Olambda__system__additive,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] : ( measur2545585918067192918nnreal @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) @ F ) ).
% sets.lambda_system_additive
thf(fact_1277_sets_Olambda__system__Compl,axiom,
! [X3: set_a,M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( member_set_a @ X3 @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) )
=> ( member_set_a @ ( minus_minus_set_a @ ( sigma_space_a @ M ) @ X3 ) @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) ) ) ).
% sets.lambda_system_Compl
% Helper facts (3)
thf(help_If_3_1_If_001tf__a_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $true @ X3 @ Y2 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
down_ray_a @ ( sup_sup_set_a @ i @ ( set_ord_atMost_a @ x ) ) ).
%------------------------------------------------------------------------------