TPTP Problem File: SLH0182^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Median_Method/0000_Median/prob_00058_002168__14631958_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1393 ( 562 unt; 115 typ; 0 def)
% Number of atoms : 3838 (1014 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11265 ( 360 ~; 79 |; 213 &;8910 @)
% ( 0 <=>;1703 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 487 ( 487 >; 0 *; 0 +; 0 <<)
% Number of symbols : 112 ( 109 usr; 9 con; 0-5 aty)
% Number of variables : 3777 ( 298 ^;3410 !; 69 ?;3777 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:43:08.006
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Filter__Ofilter_Itf__a_J_J,type,
set_filter_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Filter__Ofilter_Itf__a_J,type,
filter_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 (109)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001tf__a,type,
condit5901475214736682318elow_a: set_a > $o ).
thf(sy_c_Connected_Oconnected__component_001tf__a,type,
connec6912146525016367596nent_a: set_a > a > a > $o ).
thf(sy_c_Extended__Real__Limits_Oorder__class_Omono__set_001t__Set__Oset_Itf__a_J,type,
extend347329919781519187_set_a: set_set_a > $o ).
thf(sy_c_Extended__Real__Limits_Oorder__class_Omono__set_001tf__a,type,
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thf(sy_c_Filter_Oeventually_001tf__a,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Filter__Ofilter_Itf__a_J,type,
inf_inf_filter_a: filter_a > filter_a > filter_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,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_Osemilattice__neutr__order_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Filter__Ofilter_Itf__a_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,
sup_su2076012971530813682_set_a: set_set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001t__Filter__Ofilter_Itf__a_J,type,
measur8564814331811021273lter_a: set_set_set_a > ( set_set_a > filter_a ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001tf__a,type,
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thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Filter__Ofilter_Itf__a_J,type,
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thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Measure__Space_Oincreasing_001tf__a_001tf__a,type,
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thf(sy_c_Measure__Space_Osup__lexord_001t__Filter__Ofilter_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Measure__Space_Osup__lexord_001t__Filter__Ofilter_Itf__a_J_001tf__a,type,
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thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001tf__a,type,
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thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_Itf__a_J_001t__Filter__Ofilter_Itf__a_J,type,
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thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_Itf__a_J_001tf__a,type,
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thf(sy_c_Measure__Space_Osup__lexord_001tf__a_001t__Filter__Ofilter_Itf__a_J,type,
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thf(sy_c_Measure__Space_Osup__lexord_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Measure__Space_Osup__lexord_001tf__a_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Measure__Space_Osup__lexord_001tf__a_001tf__a,type,
measur6235662355876231836rd_a_a: a > a > ( a > a ) > a > a > a ).
thf(sy_c_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Median_Oup__ray_001tf__a,type,
up_ray_a: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_Itf__a_J,type,
bot_bot_filter_a: filter_a ).
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,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Filter__Ofilter_Itf__a_J,type,
ord_less_filter_a: filter_a > filter_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Filter__Ofilter_Itf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
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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_001t__Filter__Ofilter_Itf__a_J,type,
ord_less_eq_filter_a: filter_a > filter_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Filter__Ofilter_Itf__a_J_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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Filter__Ofilter_Itf__a_J,type,
order_3128678610506401501lter_a: ( filter_a > $o ) > filter_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
order_3565860530148683671_set_a: ( set_set_a > $o ) > set_set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
order_Greatest_a: ( a > $o ) > a ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_Itf__a_J,type,
ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Filter__Ofilter_Itf__a_J,type,
set_or9026928291982918841lter_a: filter_a > filter_a > set_filter_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or4761464488706262899_set_a: set_set_a > set_set_a > set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
set_or6288561110385358355_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001tf__a,type,
set_or672772299803893939Most_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Filter__Ofilter_Itf__a_J,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_Itf__a_J_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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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001tf__a,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Filter__Ofilter_Itf__a_J,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_Itf__a_J,type,
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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_001t__Filter__Ofilter_Itf__a_J,type,
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thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001tf__a,type,
set_ord_atMost_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Filter__Ofilter_Itf__a_J,type,
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thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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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_It__Set__Oset_Itf__a_J_J,type,
set_or5544419673604628330_set_a: set_set_a > set_set_a > set_set_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_001tf__a,type,
set_or8632414552788122084Than_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001tf__a,type,
set_ord_lessThan_a: a > set_a ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__a,type,
topolo8477419352202985285open_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001tf__a,type,
topolo1902352237885396414thin_a: a > set_a > filter_a ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oclosed_001tf__a,type,
topolo784654279908865136osed_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected_001tf__a,type,
topolo2370605967727889109cted_a: set_a > $o ).
thf(sy_c_Zorn_Ochain__subset_001t__Set__Oset_Itf__a_J,type,
chain_subset_set_a: set_set_set_a > $o ).
thf(sy_c_Zorn_Ochain__subset_001tf__a,type,
chain_subset_a: set_set_a > $o ).
thf(sy_c_Zorn_Ochains_001tf__a,type,
chains_a: set_set_a > set_set_set_a ).
thf(sy_c_member_001t__Filter__Ofilter_Itf__a_J,type,
member_filter_a: filter_a > set_filter_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (1274)
thf(fact_0_assms,axiom,
up_ray_a @ i ).
% assms
thf(fact_1_c,axiom,
member_a @ x @ i ).
% c
thf(fact_2_b,axiom,
~ ( topolo784654279908865136osed_a @ i ) ).
% b
thf(fact_3_False,axiom,
~ ? [Y: a] :
( ( ord_less_a @ Y @ x )
& ( member_a @ Y @ i ) ) ).
% False
thf(fact_4__092_060open_062I_A_092_060subseteq_062_A_123x_O_O_125_092_060close_062,axiom,
ord_less_eq_set_a @ i @ ( set_ord_atLeast_a @ x ) ).
% \<open>I \<subseteq> {x..}\<close>
thf(fact_5__092_060open_062_123x_O_O_125_A_092_060subseteq_062_AI_092_060close_062,axiom,
ord_less_eq_set_a @ ( set_ord_atLeast_a @ x ) @ i ).
% \<open>{x..} \<subseteq> I\<close>
thf(fact_6__092_060open_062I_A_061_A_123x_O_O_125_092_060close_062,axiom,
( i
= ( set_ord_atLeast_a @ x ) ) ).
% \<open>I = {x..}\<close>
thf(fact_7_subsetI,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ! [X: set_set_a] :
( ( member_set_set_a @ X @ A )
=> ( member_set_set_a @ X @ B ) )
=> ( ord_le5722252365846178494_set_a @ A @ B ) ) ).
% subsetI
thf(fact_8_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_9_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ X @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_10_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_11_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_12_order__refl,axiom,
! [X2: a] : ( ord_less_eq_a @ X2 @ X2 ) ).
% order_refl
thf(fact_13_order__refl,axiom,
! [X2: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_14_order__refl,axiom,
! [X2: filter_a] : ( ord_less_eq_filter_a @ X2 @ X2 ) ).
% order_refl
thf(fact_15_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_16_dual__order_Orefl,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_17_dual__order_Orefl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_18_dual__order_Orefl,axiom,
! [A2: filter_a] : ( ord_less_eq_filter_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_19_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_20_up__ray__def,axiom,
( up_ray_a
= ( ^ [I: set_a] :
! [X3: a,Y2: a] :
( ( member_a @ X3 @ I )
=> ( ( ord_less_eq_a @ X3 @ Y2 )
=> ( member_a @ Y2 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_21_interval__def,axiom,
( interval_a
= ( ^ [I: set_a] :
! [X3: a,Y2: a,Z: a] :
( ( member_a @ X3 @ I )
=> ( ( member_a @ Z @ I )
=> ( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z )
=> ( member_a @ Y2 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_22_topological__space__class_OopenI,axiom,
! [S: set_a] :
( ! [X: a] :
( ( member_a @ X @ S )
=> ? [T: set_a] :
( ( topolo8477419352202985285open_a @ T )
& ( member_a @ X @ T )
& ( ord_less_eq_set_a @ T @ S ) ) )
=> ( topolo8477419352202985285open_a @ S ) ) ).
% topological_space_class.openI
thf(fact_23_open__subopen,axiom,
( topolo8477419352202985285open_a
= ( ^ [S2: set_a] :
! [X3: a] :
( ( member_a @ X3 @ S2 )
=> ? [T2: set_a] :
( ( topolo8477419352202985285open_a @ T2 )
& ( member_a @ X3 @ T2 )
& ( ord_less_eq_set_a @ T2 @ S2 ) ) ) ) ) ).
% open_subopen
thf(fact_24__092_060open_062closed_A_123x_O_O_125_092_060close_062,axiom,
topolo784654279908865136osed_a @ ( set_ord_atLeast_a @ x ) ).
% \<open>closed {x..}\<close>
thf(fact_25_less__imp__neq,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_26_less__imp__neq,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_27_less__imp__neq,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_28_order_Oasym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_29_order_Oasym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_set_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_30_order_Oasym,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ~ ( ord_less_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_31_ord__eq__less__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_32_ord__eq__less__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A2 = B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_33_ord__eq__less__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_34_ord__less__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_35_ord__less__eq__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_36_ord__less__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_37_antisym__conv3,axiom,
! [Y3: a,X2: a] :
( ~ ( ord_less_a @ Y3 @ X2 )
=> ( ( ~ ( ord_less_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_38_linorder__cases,axiom,
! [X2: a,Y3: a] :
( ~ ( ord_less_a @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_a @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_39_dual__order_Oasym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_40_dual__order_Oasym,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_set_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_41_dual__order_Oasym,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ~ ( ord_less_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_42_dual__order_Oirrefl,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_43_dual__order_Oirrefl,axiom,
! [A2: set_set_a] :
~ ( ord_less_set_set_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_44_dual__order_Oirrefl,axiom,
! [A2: a] :
~ ( ord_less_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_45_linorder__less__wlog,axiom,
! [P: a > a > $o,A2: a,B2: a] :
( ! [A3: a,B3: a] :
( ( ord_less_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a] : ( P @ A3 @ A3 )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_46_order_Ostrict__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_47_order_Ostrict__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_48_order_Ostrict__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_49_not__less__iff__gr__or__eq,axiom,
! [X2: a,Y3: a] :
( ( ~ ( ord_less_a @ X2 @ Y3 ) )
= ( ( ord_less_a @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_50_dual__order_Ostrict__trans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_51_dual__order_Ostrict__trans,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ( ord_less_set_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_52_dual__order_Ostrict__trans,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ( ord_less_a @ C @ B2 )
=> ( ord_less_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_53_order_Ostrict__implies__not__eq,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_54_order_Ostrict__implies__not__eq,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_55_order_Ostrict__implies__not__eq,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_56_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_57_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_58_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_59_linorder__neqE,axiom,
! [X2: a,Y3: a] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_a @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_60_order__less__asym,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_61_order__less__asym,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y3 )
=> ~ ( ord_less_set_set_a @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_62_order__less__asym,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_63_linorder__neq__iff,axiom,
! [X2: a,Y3: a] :
( ( X2 != Y3 )
= ( ( ord_less_a @ X2 @ Y3 )
| ( ord_less_a @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_64_order__less__asym_H,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_65_order__less__asym_H,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_set_a @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_66_order__less__asym_H,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ~ ( ord_less_a @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_67_order__less__trans,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_68_order__less__trans,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_set_a @ Y3 @ Z2 )
=> ( ord_less_set_set_a @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_69_order__less__trans,axiom,
! [X2: a,Y3: a,Z2: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ( ord_less_a @ Y3 @ Z2 )
=> ( ord_less_a @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_70_ord__eq__less__subst,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_71_ord__eq__less__subst,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_72_ord__eq__less__subst,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_73_ord__eq__less__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_74_ord__eq__less__subst,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_75_ord__eq__less__subst,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_76_ord__eq__less__subst,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_77_ord__eq__less__subst,axiom,
! [A2: set_set_a,F: set_set_a > set_set_a,B2: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_78_ord__eq__less__subst,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_79_ord__less__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_80_ord__less__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_81_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_82_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_83_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_84_ord__less__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_85_ord__less__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_86_ord__less__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_87_ord__less__eq__subst,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_88_order__less__irrefl,axiom,
! [X2: set_a] :
~ ( ord_less_set_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_89_order__less__irrefl,axiom,
! [X2: set_set_a] :
~ ( ord_less_set_set_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_90_order__less__irrefl,axiom,
! [X2: a] :
~ ( ord_less_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_91_order__less__subst1,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_92_order__less__subst1,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_93_order__less__subst1,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_94_order__less__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_95_order__less__subst1,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_96_order__less__subst1,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( ord_less_set_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_97_order__less__subst1,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_98_order__less__subst1,axiom,
! [A2: set_set_a,F: set_set_a > set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_99_order__less__subst1,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_100_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_101_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_102_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_103_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_104_Collect__mem__eq,axiom,
! [A: set_set_set_a] :
( ( collect_set_set_a
@ ^ [X3: set_set_a] : ( member_set_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_105_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_106_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_107_order__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_108_order__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_109_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_110_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_111_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_112_order__less__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_113_order__less__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_114_order__less__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_115_order__less__subst2,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_116_order__less__not__sym,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_117_order__less__not__sym,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y3 )
=> ~ ( ord_less_set_set_a @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_118_order__less__not__sym,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_119_order__less__imp__triv,axiom,
! [X2: set_a,Y3: set_a,P: $o] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_120_order__less__imp__triv,axiom,
! [X2: set_set_a,Y3: set_set_a,P: $o] :
( ( ord_less_set_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_set_a @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_121_order__less__imp__triv,axiom,
! [X2: a,Y3: a,P: $o] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ( ord_less_a @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_122_linorder__less__linear,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_a @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_123_order__less__imp__not__eq,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_124_order__less__imp__not__eq,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_125_order__less__imp__not__eq,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_126_order__less__imp__not__eq2,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_127_order__less__imp__not__eq2,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_128_order__less__imp__not__eq2,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_129_order__less__imp__not__less,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_130_order__less__imp__not__less,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y3 )
=> ~ ( ord_less_set_set_a @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_131_order__less__imp__not__less,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_132_closed__atLeast,axiom,
! [A2: a] : ( topolo784654279908865136osed_a @ ( set_ord_atLeast_a @ A2 ) ) ).
% closed_atLeast
thf(fact_133_order__le__imp__less__or__eq,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_set_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_134_order__le__imp__less__or__eq,axiom,
! [X2: filter_a,Y3: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ Y3 )
=> ( ( ord_less_filter_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_135_order__le__imp__less__or__eq,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_136_order__le__imp__less__or__eq,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_137_linorder__le__less__linear,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
| ( ord_less_a @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_138_order__less__le__subst2,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_139_order__less__le__subst2,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_140_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_141_order__less__le__subst2,axiom,
! [A2: a,B2: a,F: a > filter_a,C: filter_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_filter_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_filter_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_142_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_143_order__less__le__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_144_order__less__le__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_145_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > filter_a,C: filter_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_filter_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_filter_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_146_order__less__le__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_147_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_148_order__less__le__subst1,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_149_order__less__le__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_150_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 )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_151_order__less__le__subst1,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_152_order__less__le__subst1,axiom,
! [A2: filter_a,F: a > filter_a,B2: a,C: a] :
( ( ord_less_filter_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_filter_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_153_order__less__le__subst1,axiom,
! [A2: a,F: filter_a > a,B2: filter_a,C: filter_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_filter_a @ B2 @ C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_154_order__less__le__subst1,axiom,
! [A2: filter_a,F: set_a > filter_a,B2: set_a,C: set_a] :
( ( ord_less_filter_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_filter_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_155_order__less__le__subst1,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( ord_less_set_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_156_order__less__le__subst1,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_157_order__less__le__subst1,axiom,
! [A2: set_a,F: filter_a > set_a,B2: filter_a,C: filter_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_filter_a @ B2 @ C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_158_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_159_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_160_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 )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_161_order__le__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_162_order__le__less__subst2,axiom,
! [A2: a,B2: a,F: a > filter_a,C: filter_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_filter_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_filter_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_163_order__le__less__subst2,axiom,
! [A2: filter_a,B2: filter_a,F: filter_a > a,C: a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_164_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > filter_a,C: filter_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_filter_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_filter_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_165_order__le__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_166_order__le__less__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_167_order__le__less__subst2,axiom,
! [A2: filter_a,B2: filter_a,F: filter_a > set_a,C: set_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_168_order__le__less__subst1,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_169_order__le__less__subst1,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_170_order__le__less__subst1,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_171_order__le__less__subst1,axiom,
! [A2: filter_a,F: a > filter_a,B2: a,C: a] :
( ( ord_less_eq_filter_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_filter_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_172_order__le__less__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_173_order__le__less__subst1,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_174_order__le__less__subst1,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_175_order__le__less__subst1,axiom,
! [A2: filter_a,F: set_a > filter_a,B2: set_a,C: set_a] :
( ( ord_less_eq_filter_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_filter_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_176_order__le__less__subst1,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X @ Y4 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_177_order__le__less__subst1,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_178_order__less__le__trans,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ Z2 )
=> ( ord_less_set_set_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_179_order__less__le__trans,axiom,
! [X2: filter_a,Y3: filter_a,Z2: filter_a] :
( ( ord_less_filter_a @ X2 @ Y3 )
=> ( ( ord_less_eq_filter_a @ Y3 @ Z2 )
=> ( ord_less_filter_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_180_order__less__le__trans,axiom,
! [X2: a,Y3: a,Z2: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ Z2 )
=> ( ord_less_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_181_order__less__le__trans,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_182_order__le__less__trans,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_set_a @ Y3 @ Z2 )
=> ( ord_less_set_set_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_183_order__le__less__trans,axiom,
! [X2: filter_a,Y3: filter_a,Z2: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ Y3 )
=> ( ( ord_less_filter_a @ Y3 @ Z2 )
=> ( ord_less_filter_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_184_order__le__less__trans,axiom,
! [X2: a,Y3: a,Z2: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_a @ Y3 @ Z2 )
=> ( ord_less_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_185_order__le__less__trans,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_186_order__neq__le__trans,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 != B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_187_order__neq__le__trans,axiom,
! [A2: filter_a,B2: filter_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ord_less_filter_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_188_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_189_order__neq__le__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_190_order__le__neq__trans,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_191_order__le__neq__trans,axiom,
! [A2: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_filter_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_192_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_193_order__le__neq__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_194_order__less__imp__le,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_195_order__less__imp__le,axiom,
! [X2: filter_a,Y3: filter_a] :
( ( ord_less_filter_a @ X2 @ Y3 )
=> ( ord_less_eq_filter_a @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_196_order__less__imp__le,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_197_order__less__imp__le,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_198_linorder__not__less,axiom,
! [X2: a,Y3: a] :
( ( ~ ( ord_less_a @ X2 @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_199_linorder__not__le,axiom,
! [X2: a,Y3: a] :
( ( ~ ( ord_less_eq_a @ X2 @ Y3 ) )
= ( ord_less_a @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_200_order__less__le,axiom,
( ord_less_set_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_201_order__less__le,axiom,
( ord_less_filter_a
= ( ^ [X3: filter_a,Y2: filter_a] :
( ( ord_less_eq_filter_a @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_202_order__less__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_203_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_204_order__le__less,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] :
( ( ord_less_set_set_a @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_205_order__le__less,axiom,
( ord_less_eq_filter_a
= ( ^ [X3: filter_a,Y2: filter_a] :
( ( ord_less_filter_a @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_206_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X3: a,Y2: a] :
( ( ord_less_a @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_207_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_208_dual__order_Ostrict__implies__order,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_209_dual__order_Ostrict__implies__order,axiom,
! [B2: filter_a,A2: filter_a] :
( ( ord_less_filter_a @ B2 @ A2 )
=> ( ord_less_eq_filter_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_210_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_211_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_212_order_Ostrict__implies__order,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_213_order_Ostrict__implies__order,axiom,
! [A2: filter_a,B2: filter_a] :
( ( ord_less_filter_a @ A2 @ B2 )
=> ( ord_less_eq_filter_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_214_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_215_order_Ostrict__implies__order,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_216_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ~ ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_217_dual__order_Ostrict__iff__not,axiom,
( ord_less_filter_a
= ( ^ [B4: filter_a,A4: filter_a] :
( ( ord_less_eq_filter_a @ B4 @ A4 )
& ~ ( ord_less_eq_filter_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_218_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ~ ( ord_less_eq_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_219_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_220_dual__order_Ostrict__trans2,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_221_dual__order_Ostrict__trans2,axiom,
! [B2: filter_a,A2: filter_a,C: filter_a] :
( ( ord_less_filter_a @ B2 @ A2 )
=> ( ( ord_less_eq_filter_a @ C @ B2 )
=> ( ord_less_filter_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_222_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_223_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_224_dual__order_Ostrict__trans1,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_less_set_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_225_dual__order_Ostrict__trans1,axiom,
! [B2: filter_a,A2: filter_a,C: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ A2 )
=> ( ( ord_less_filter_a @ C @ B2 )
=> ( ord_less_filter_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_226_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_227_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_228_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_229_dual__order_Ostrict__iff__order,axiom,
( ord_less_filter_a
= ( ^ [B4: filter_a,A4: filter_a] :
( ( ord_less_eq_filter_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_230_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_231_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_232_dual__order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_less_set_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_233_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_filter_a
= ( ^ [B4: filter_a,A4: filter_a] :
( ( ord_less_filter_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_234_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_235_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_236_order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ~ ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_237_order_Ostrict__iff__not,axiom,
( ord_less_filter_a
= ( ^ [A4: filter_a,B4: filter_a] :
( ( ord_less_eq_filter_a @ A4 @ B4 )
& ~ ( ord_less_eq_filter_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_238_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ~ ( ord_less_eq_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_239_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_240_order_Ostrict__trans2,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_241_order_Ostrict__trans2,axiom,
! [A2: filter_a,B2: filter_a,C: filter_a] :
( ( ord_less_filter_a @ A2 @ B2 )
=> ( ( ord_less_eq_filter_a @ B2 @ C )
=> ( ord_less_filter_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_242_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_243_order_Ostrict__trans2,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_244_order_Ostrict__trans1,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_245_order_Ostrict__trans1,axiom,
! [A2: filter_a,B2: filter_a,C: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( ord_less_filter_a @ B2 @ C )
=> ( ord_less_filter_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_246_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_247_order_Ostrict__trans1,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_248_order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_249_order_Ostrict__iff__order,axiom,
( ord_less_filter_a
= ( ^ [A4: filter_a,B4: filter_a] :
( ( ord_less_eq_filter_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_250_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_251_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_252_order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_less_set_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_253_order_Oorder__iff__strict,axiom,
( ord_less_eq_filter_a
= ( ^ [A4: filter_a,B4: filter_a] :
( ( ord_less_filter_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_254_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_255_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_256_not__le__imp__less,axiom,
! [Y3: a,X2: a] :
( ~ ( ord_less_eq_a @ Y3 @ X2 )
=> ( ord_less_a @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_257_less__le__not__le,axiom,
( ord_less_set_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
& ~ ( ord_le3724670747650509150_set_a @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_258_less__le__not__le,axiom,
( ord_less_filter_a
= ( ^ [X3: filter_a,Y2: filter_a] :
( ( ord_less_eq_filter_a @ X3 @ Y2 )
& ~ ( ord_less_eq_filter_a @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_259_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
& ~ ( ord_less_eq_a @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_260_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
& ~ ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_261_antisym__conv2,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_set_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_262_antisym__conv2,axiom,
! [X2: filter_a,Y3: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ Y3 )
=> ( ( ~ ( ord_less_filter_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_263_antisym__conv2,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ~ ( ord_less_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_264_antisym__conv2,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_265_antisym__conv1,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ~ ( ord_less_set_set_a @ X2 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_266_antisym__conv1,axiom,
! [X2: filter_a,Y3: filter_a] :
( ~ ( ord_less_filter_a @ X2 @ Y3 )
=> ( ( ord_less_eq_filter_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_267_antisym__conv1,axiom,
! [X2: a,Y3: a] :
( ~ ( ord_less_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_268_antisym__conv1,axiom,
! [X2: set_a,Y3: set_a] :
( ~ ( ord_less_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_269_nless__le,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ~ ( ord_less_set_set_a @ A2 @ B2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_270_nless__le,axiom,
! [A2: filter_a,B2: filter_a] :
( ( ~ ( ord_less_filter_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_filter_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_271_nless__le,axiom,
! [A2: a,B2: a] :
( ( ~ ( ord_less_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_272_nless__le,axiom,
! [A2: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_273_leI,axiom,
! [X2: a,Y3: a] :
( ~ ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% leI
thf(fact_274_leD,axiom,
! [Y3: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X2 )
=> ~ ( ord_less_set_set_a @ X2 @ Y3 ) ) ).
% leD
thf(fact_275_leD,axiom,
! [Y3: filter_a,X2: filter_a] :
( ( ord_less_eq_filter_a @ Y3 @ X2 )
=> ~ ( ord_less_filter_a @ X2 @ Y3 ) ) ).
% leD
thf(fact_276_leD,axiom,
! [Y3: a,X2: a] :
( ( ord_less_eq_a @ Y3 @ X2 )
=> ~ ( ord_less_a @ X2 @ Y3 ) ) ).
% leD
thf(fact_277_leD,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ~ ( ord_less_set_a @ X2 @ Y3 ) ) ).
% leD
thf(fact_278_order__antisym__conv,axiom,
! [Y3: a,X2: a] :
( ( ord_less_eq_a @ Y3 @ X2 )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_279_order__antisym__conv,axiom,
! [Y3: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_280_order__antisym__conv,axiom,
! [Y3: filter_a,X2: filter_a] :
( ( ord_less_eq_filter_a @ Y3 @ X2 )
=> ( ( ord_less_eq_filter_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_281_order__antisym__conv,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_282_linorder__le__cases,axiom,
! [X2: a,Y3: a] :
( ~ ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_283_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_284_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_285_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 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_286_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 )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_287_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > filter_a,C: filter_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_filter_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_288_ord__le__eq__subst,axiom,
! [A2: filter_a,B2: filter_a,F: filter_a > a,C: a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_289_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > filter_a,C: filter_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_filter_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_290_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_291_ord__le__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_292_ord__le__eq__subst,axiom,
! [A2: filter_a,B2: filter_a,F: filter_a > set_a,C: set_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_293_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_294_ord__eq__le__subst,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_295_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 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_296_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 )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_297_ord__eq__le__subst,axiom,
! [A2: filter_a,F: a > filter_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_filter_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_298_ord__eq__le__subst,axiom,
! [A2: a,F: filter_a > a,B2: filter_a,C: filter_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_filter_a @ B2 @ C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_299_ord__eq__le__subst,axiom,
! [A2: filter_a,F: set_a > filter_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_filter_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_300_ord__eq__le__subst,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_301_ord__eq__le__subst,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_302_ord__eq__le__subst,axiom,
! [A2: set_a,F: filter_a > set_a,B2: filter_a,C: filter_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_filter_a @ B2 @ C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_303_linorder__linear,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
| ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_304_order__eq__refl,axiom,
! [X2: a,Y3: a] :
( ( X2 = Y3 )
=> ( ord_less_eq_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_305_order__eq__refl,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( X2 = Y3 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_306_order__eq__refl,axiom,
! [X2: filter_a,Y3: filter_a] :
( ( X2 = Y3 )
=> ( ord_less_eq_filter_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_307_order__eq__refl,axiom,
! [X2: set_a,Y3: set_a] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_308_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_309_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 )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_310_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 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_311_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 )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_312_order__subst2,axiom,
! [A2: a,B2: a,F: a > filter_a,C: filter_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_filter_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_filter_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_313_order__subst2,axiom,
! [A2: filter_a,B2: filter_a,F: filter_a > a,C: a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_314_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > filter_a,C: filter_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_filter_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_filter_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_315_order__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_316_order__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_317_order__subst2,axiom,
! [A2: filter_a,B2: filter_a,F: filter_a > set_a,C: set_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_318_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_319_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 )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_320_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 )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_321_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 )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_322_order__subst1,axiom,
! [A2: a,F: filter_a > a,B2: filter_a,C: filter_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_filter_a @ B2 @ C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_323_order__subst1,axiom,
! [A2: filter_a,F: a > filter_a,B2: a,C: a] :
( ( ord_less_eq_filter_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_filter_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_324_order__subst1,axiom,
! [A2: set_a,F: filter_a > set_a,B2: filter_a,C: filter_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_filter_a @ B2 @ C )
=> ( ! [X: filter_a,Y4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_325_order__subst1,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y4 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_326_order__subst1,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X: a,Y4: a] :
( ( ord_less_eq_a @ X @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_327_order__subst1,axiom,
! [A2: filter_a,F: set_a > filter_a,B2: set_a,C: set_a] :
( ( ord_less_eq_filter_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ord_less_eq_filter_a @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_filter_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_328_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: a,Z3: a] : ( Y5 = Z3 ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ( ord_less_eq_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_329_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_330_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: filter_a,Z3: filter_a] : ( Y5 = Z3 ) )
= ( ^ [A4: filter_a,B4: filter_a] :
( ( ord_less_eq_filter_a @ A4 @ B4 )
& ( ord_less_eq_filter_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_331_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_332_antisym,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_333_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_334_antisym,axiom,
! [A2: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( ord_less_eq_filter_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_335_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_336_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_337_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_338_dual__order_Otrans,axiom,
! [B2: filter_a,A2: filter_a,C: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ A2 )
=> ( ( ord_less_eq_filter_a @ C @ B2 )
=> ( ord_less_eq_filter_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_339_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_340_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_341_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_342_dual__order_Oantisym,axiom,
! [B2: filter_a,A2: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ A2 )
=> ( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_343_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_344_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: a,Z3: a] : ( Y5 = Z3 ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ( ord_less_eq_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_345_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_346_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: filter_a,Z3: filter_a] : ( Y5 = Z3 ) )
= ( ^ [A4: filter_a,B4: filter_a] :
( ( ord_less_eq_filter_a @ B4 @ A4 )
& ( ord_less_eq_filter_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_347_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_348_linorder__wlog,axiom,
! [P: a > a > $o,A2: a,B2: a] :
( ! [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_349_order__trans,axiom,
! [X2: a,Y3: a,Z2: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ Z2 )
=> ( ord_less_eq_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_350_order__trans,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_351_order__trans,axiom,
! [X2: filter_a,Y3: filter_a,Z2: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ Y3 )
=> ( ( ord_less_eq_filter_a @ Y3 @ Z2 )
=> ( ord_less_eq_filter_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_352_order__trans,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z2 )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_353_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_354_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_355_order_Otrans,axiom,
! [A2: filter_a,B2: filter_a,C: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( ord_less_eq_filter_a @ B2 @ C )
=> ( ord_less_eq_filter_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_356_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_357_order__antisym,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_358_order__antisym,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_359_order__antisym,axiom,
! [X2: filter_a,Y3: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ Y3 )
=> ( ( ord_less_eq_filter_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_360_order__antisym,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_361_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_362_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_363_ord__le__eq__trans,axiom,
! [A2: filter_a,B2: filter_a,C: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_filter_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_364_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_365_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_366_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_367_ord__eq__le__trans,axiom,
! [A2: filter_a,B2: filter_a,C: filter_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_filter_a @ B2 @ C )
=> ( ord_less_eq_filter_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_368_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_369_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: a,Z3: a] : ( Y5 = Z3 ) )
= ( ^ [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
& ( ord_less_eq_a @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_370_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
& ( ord_le3724670747650509150_set_a @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_371_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: filter_a,Z3: filter_a] : ( Y5 = Z3 ) )
= ( ^ [X3: filter_a,Y2: filter_a] :
( ( ord_less_eq_filter_a @ X3 @ Y2 )
& ( ord_less_eq_filter_a @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_372_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_373_le__cases3,axiom,
! [X2: a,Y3: a,Z2: a] :
( ( ( ord_less_eq_a @ X2 @ Y3 )
=> ~ ( ord_less_eq_a @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_a @ Y3 @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_a @ X2 @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_a @ Z2 @ Y3 )
=> ~ ( ord_less_eq_a @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_a @ Y3 @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_a @ Z2 @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_374_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_375_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_376_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_377_set__eq__subset,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_378_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_379_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_380_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_381_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_382_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_383_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_384_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_385_subset__iff,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A5: set_set_set_a,B5: set_set_set_a] :
! [T3: set_set_a] :
( ( member_set_set_a @ T3 @ A5 )
=> ( member_set_set_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_386_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T3: set_a] :
( ( member_set_a @ T3 @ A5 )
=> ( member_set_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_387_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A5 )
=> ( member_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_388_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_389_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_390_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_391_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_392_subset__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A5: set_set_set_a,B5: set_set_set_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A5 )
=> ( member_set_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_393_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( member_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_394_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_395_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_396_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_397_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_398_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_399_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_400_in__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a,X2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( member_set_set_a @ X2 @ A )
=> ( member_set_set_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_401_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_402_in__mono,axiom,
! [A: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_403_separation__t1,axiom,
! [X2: a,Y3: a] :
( ( X2 != Y3 )
= ( ? [U: set_a] :
( ( topolo8477419352202985285open_a @ U )
& ( member_a @ X2 @ U )
& ~ ( member_a @ Y3 @ U ) ) ) ) ).
% separation_t1
thf(fact_404_separation__t0,axiom,
! [X2: a,Y3: a] :
( ( X2 != Y3 )
= ( ? [U: set_a] :
( ( topolo8477419352202985285open_a @ U )
& ( ( member_a @ X2 @ U )
!= ( member_a @ Y3 @ U ) ) ) ) ) ).
% separation_t0
thf(fact_405_t1__space,axiom,
! [X2: a,Y3: a] :
( ( X2 != Y3 )
=> ? [U2: set_a] :
( ( topolo8477419352202985285open_a @ U2 )
& ( member_a @ X2 @ U2 )
& ~ ( member_a @ Y3 @ U2 ) ) ) ).
% t1_space
thf(fact_406_t0__space,axiom,
! [X2: a,Y3: a] :
( ( X2 != Y3 )
=> ? [U2: set_a] :
( ( topolo8477419352202985285open_a @ U2 )
& ( ( member_a @ X2 @ U2 )
!= ( member_a @ Y3 @ U2 ) ) ) ) ).
% t0_space
thf(fact_407_atLeast__subset__iff,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or3904034815786525833_set_a @ X2 ) @ ( set_or3904034815786525833_set_a @ Y3 ) )
= ( ord_le3724670747650509150_set_a @ Y3 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_408_atLeast__subset__iff,axiom,
! [X2: filter_a,Y3: filter_a] :
( ( ord_le8471427362357671172lter_a @ ( set_or1258606792136358095lter_a @ X2 ) @ ( set_or1258606792136358095lter_a @ Y3 ) )
= ( ord_less_eq_filter_a @ Y3 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_409_atLeast__subset__iff,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or8362275514725411625_set_a @ X2 ) @ ( set_or8362275514725411625_set_a @ Y3 ) )
= ( ord_less_eq_set_a @ Y3 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_410_atLeast__subset__iff,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ X2 ) @ ( set_ord_atLeast_a @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_411_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_412_atLeast__iff,axiom,
! [I2: filter_a,K: filter_a] :
( ( member_filter_a @ I2 @ ( set_or1258606792136358095lter_a @ K ) )
= ( ord_less_eq_filter_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_413_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_414_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_415_atLeast__eq__iff,axiom,
! [X2: set_a,Y3: set_a] :
( ( ( set_or8362275514725411625_set_a @ X2 )
= ( set_or8362275514725411625_set_a @ Y3 ) )
= ( X2 = Y3 ) ) ).
% atLeast_eq_iff
thf(fact_416_atLeast__eq__iff,axiom,
! [X2: a,Y3: a] :
( ( ( set_ord_atLeast_a @ X2 )
= ( set_ord_atLeast_a @ Y3 ) )
= ( X2 = Y3 ) ) ).
% atLeast_eq_iff
thf(fact_417_verit__comp__simplify1_I3_J,axiom,
! [B6: a,A6: a] :
( ( ~ ( ord_less_eq_a @ B6 @ A6 ) )
= ( ord_less_a @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_418_pinf_I6_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ~ ( ord_less_eq_a @ X4 @ T4 ) ) ).
% pinf(6)
thf(fact_419_pinf_I8_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( ord_less_eq_a @ T4 @ X4 ) ) ).
% pinf(8)
thf(fact_420_minf_I6_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( ord_less_eq_a @ X4 @ T4 ) ) ).
% minf(6)
thf(fact_421_minf_I8_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ~ ( ord_less_eq_a @ T4 @ X4 ) ) ).
% minf(8)
thf(fact_422_Greatest__equality,axiom,
! [P: a > $o,X2: a] :
( ( P @ X2 )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X2 ) )
=> ( ( order_Greatest_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_423_Greatest__equality,axiom,
! [P: set_set_a > $o,X2: set_set_a] :
( ( P @ X2 )
=> ( ! [Y4: set_set_a] :
( ( P @ Y4 )
=> ( ord_le3724670747650509150_set_a @ Y4 @ X2 ) )
=> ( ( order_3565860530148683671_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_424_Greatest__equality,axiom,
! [P: filter_a > $o,X2: filter_a] :
( ( P @ X2 )
=> ( ! [Y4: filter_a] :
( ( P @ Y4 )
=> ( ord_less_eq_filter_a @ Y4 @ X2 ) )
=> ( ( order_3128678610506401501lter_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_425_Greatest__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y4: set_a] :
( ( P @ Y4 )
=> ( ord_less_eq_set_a @ Y4 @ X2 ) )
=> ( ( order_Greatest_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_426_psubsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_427_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_428_psubsetE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_429_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_430_psubset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_431_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_432_psubset__imp__subset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_433_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_434_psubset__subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_less_set_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_435_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_436_subset__not__subset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ~ ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_437_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_438_subset__psubset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_set_set_a @ B @ C2 )
=> ( ord_less_set_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_439_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_440_subset__iff__psubset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_less_set_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_441_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_442_verit__comp__simplify1_I2_J,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_443_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_444_verit__comp__simplify1_I2_J,axiom,
! [A2: filter_a] : ( ord_less_eq_filter_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_445_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_446_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_447_verit__comp__simplify1_I1_J,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_448_verit__comp__simplify1_I1_J,axiom,
! [A2: set_set_a] :
~ ( ord_less_set_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_449_verit__comp__simplify1_I1_J,axiom,
! [A2: a] :
~ ( ord_less_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_450_pinf_I1_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X: a] :
( ( ord_less_a @ Z5 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: a] :
! [X: a] :
( ( ord_less_a @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_451_pinf_I2_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X: a] :
( ( ord_less_a @ Z5 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: a] :
! [X: a] :
( ( ord_less_a @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_452_pinf_I3_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( X4 != T4 ) ) ).
% pinf(3)
thf(fact_453_pinf_I4_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( X4 != T4 ) ) ).
% pinf(4)
thf(fact_454_pinf_I5_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ~ ( ord_less_a @ X4 @ T4 ) ) ).
% pinf(5)
thf(fact_455_pinf_I7_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( ord_less_a @ T4 @ X4 ) ) ).
% pinf(7)
thf(fact_456_minf_I1_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X: a] :
( ( ord_less_a @ X @ Z5 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: a] :
! [X: a] :
( ( ord_less_a @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_457_minf_I2_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X: a] :
( ( ord_less_a @ X @ Z5 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: a] :
! [X: a] :
( ( ord_less_a @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_458_minf_I3_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( X4 != T4 ) ) ).
% minf(3)
thf(fact_459_minf_I4_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( X4 != T4 ) ) ).
% minf(4)
thf(fact_460_minf_I5_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( ord_less_a @ X4 @ T4 ) ) ).
% minf(5)
thf(fact_461_minf_I7_J,axiom,
! [T4: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ~ ( ord_less_a @ T4 @ X4 ) ) ).
% minf(7)
thf(fact_462_GreatestI2__order,axiom,
! [P: a > $o,X2: a,Q: a > $o] :
( ( P @ X2 )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X2 ) )
=> ( ! [X: a] :
( ( P @ X )
=> ( ! [Y: a] :
( ( P @ Y )
=> ( ord_less_eq_a @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_463_GreatestI2__order,axiom,
! [P: set_set_a > $o,X2: set_set_a,Q: set_set_a > $o] :
( ( P @ X2 )
=> ( ! [Y4: set_set_a] :
( ( P @ Y4 )
=> ( ord_le3724670747650509150_set_a @ Y4 @ X2 ) )
=> ( ! [X: set_set_a] :
( ( P @ X )
=> ( ! [Y: set_set_a] :
( ( P @ Y )
=> ( ord_le3724670747650509150_set_a @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_3565860530148683671_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_464_GreatestI2__order,axiom,
! [P: filter_a > $o,X2: filter_a,Q: filter_a > $o] :
( ( P @ X2 )
=> ( ! [Y4: filter_a] :
( ( P @ Y4 )
=> ( ord_less_eq_filter_a @ Y4 @ X2 ) )
=> ( ! [X: filter_a] :
( ( P @ X )
=> ( ! [Y: filter_a] :
( ( P @ Y )
=> ( ord_less_eq_filter_a @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_3128678610506401501lter_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_465_GreatestI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y4: set_a] :
( ( P @ Y4 )
=> ( ord_less_eq_set_a @ Y4 @ X2 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y: set_a] :
( ( P @ Y )
=> ( ord_less_eq_set_a @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_466_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_467_Icc__subset__Ici__iff,axiom,
! [L: filter_a,H: filter_a,L2: filter_a] :
( ( ord_le8471427362357671172lter_a @ ( set_or9026928291982918841lter_a @ L @ H ) @ ( set_or1258606792136358095lter_a @ L2 ) )
= ( ~ ( ord_less_eq_filter_a @ L @ H )
| ( ord_less_eq_filter_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_468_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_469_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_470_increasingD,axiom,
! [M: set_set_a,F: set_a > a,X2: set_a,Y3: set_a] :
( ( measur632008712001695539ng_a_a @ M @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( member_set_a @ X2 @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_471_increasingD,axiom,
! [M: set_set_a,F: set_a > set_set_a,X2: set_a,Y3: set_a] :
( ( measur8202069185322079731_set_a @ M @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( member_set_a @ X2 @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_472_increasingD,axiom,
! [M: set_set_a,F: set_a > filter_a,X2: set_a,Y3: set_a] :
( ( measur4328814160354775865lter_a @ M @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( member_set_a @ X2 @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_less_eq_filter_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_473_increasingD,axiom,
! [M: set_set_set_a,F: set_set_a > set_a,X2: set_set_a,Y3: set_set_a] :
( ( measur5181028491126448947_set_a @ M @ F )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( member_set_set_a @ X2 @ M )
=> ( ( member_set_set_a @ Y3 @ M )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_474_increasingD,axiom,
! [M: set_set_set_a,F: set_set_a > a,X2: set_set_a,Y3: set_set_a] :
( ( measur8059081411538669011et_a_a @ M @ F )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( member_set_set_a @ X2 @ M )
=> ( ( member_set_set_a @ Y3 @ M )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_475_increasingD,axiom,
! [M: set_set_set_a,F: set_set_a > set_set_a,X2: set_set_a,Y3: set_set_a] :
( ( measur2197171192767378579_set_a @ M @ F )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( member_set_set_a @ X2 @ M )
=> ( ( member_set_set_a @ Y3 @ M )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_476_increasingD,axiom,
! [M: set_set_set_a,F: set_set_a > filter_a,X2: set_set_a,Y3: set_set_a] :
( ( measur8564814331811021273lter_a @ M @ F )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( member_set_set_a @ X2 @ M )
=> ( ( member_set_set_a @ Y3 @ M )
=> ( ord_less_eq_filter_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_477_increasingD,axiom,
! [M: set_set_a,F: set_a > set_a,X2: set_a,Y3: set_a] :
( ( measur7842569353079325843_set_a @ M @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( member_set_a @ X2 @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_478_increasing__def,axiom,
( measur632008712001695539ng_a_a
= ( ^ [M2: set_set_a,Mu: set_a > a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ M2 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ M2 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_a @ ( Mu @ X3 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_479_increasing__def,axiom,
( measur8202069185322079731_set_a
= ( ^ [M2: set_set_a,Mu: set_a > set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ M2 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ M2 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( Mu @ X3 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_480_increasing__def,axiom,
( measur4328814160354775865lter_a
= ( ^ [M2: set_set_a,Mu: set_a > filter_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ M2 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ M2 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_filter_a @ ( Mu @ X3 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_481_increasing__def,axiom,
( measur5181028491126448947_set_a
= ( ^ [M2: set_set_set_a,Mu: set_set_a > set_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ M2 )
=> ! [Y2: set_set_a] :
( ( member_set_set_a @ Y2 @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( Mu @ X3 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_482_increasing__def,axiom,
( measur8059081411538669011et_a_a
= ( ^ [M2: set_set_set_a,Mu: set_set_a > a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ M2 )
=> ! [Y2: set_set_a] :
( ( member_set_set_a @ Y2 @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_a @ ( Mu @ X3 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_483_increasing__def,axiom,
( measur2197171192767378579_set_a
= ( ^ [M2: set_set_set_a,Mu: set_set_a > set_set_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ M2 )
=> ! [Y2: set_set_a] :
( ( member_set_set_a @ Y2 @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( Mu @ X3 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_484_increasing__def,axiom,
( measur8564814331811021273lter_a
= ( ^ [M2: set_set_set_a,Mu: set_set_a > filter_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ M2 )
=> ! [Y2: set_set_a] :
( ( member_set_set_a @ Y2 @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_filter_a @ ( Mu @ X3 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_485_increasing__def,axiom,
( measur7842569353079325843_set_a
= ( ^ [M2: set_set_a,Mu: set_a > set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ M2 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ M2 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( Mu @ X3 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_486_atLeastAtMost__iff,axiom,
! [I2: set_set_a,L: set_set_a,U3: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or4761464488706262899_set_a @ L @ U3 ) )
= ( ( ord_le3724670747650509150_set_a @ L @ I2 )
& ( ord_le3724670747650509150_set_a @ I2 @ U3 ) ) ) ).
% atLeastAtMost_iff
thf(fact_487_atLeastAtMost__iff,axiom,
! [I2: filter_a,L: filter_a,U3: filter_a] :
( ( member_filter_a @ I2 @ ( set_or9026928291982918841lter_a @ L @ U3 ) )
= ( ( ord_less_eq_filter_a @ L @ I2 )
& ( ord_less_eq_filter_a @ I2 @ U3 ) ) ) ).
% atLeastAtMost_iff
thf(fact_488_atLeastAtMost__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or672772299803893939Most_a @ L @ U3 ) )
= ( ( ord_less_eq_a @ L @ I2 )
& ( ord_less_eq_a @ I2 @ U3 ) ) ) ).
% atLeastAtMost_iff
thf(fact_489_atLeastAtMost__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or6288561110385358355_set_a @ L @ U3 ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U3 ) ) ) ).
% atLeastAtMost_iff
thf(fact_490_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_491_Icc__eq__Icc,axiom,
! [L: filter_a,H: filter_a,L2: filter_a,H2: filter_a] :
( ( ( set_or9026928291982918841lter_a @ L @ H )
= ( set_or9026928291982918841lter_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_filter_a @ L @ H )
& ~ ( ord_less_eq_filter_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_492_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_493_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_494_atLeastatMost__subset__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a,D: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ A2 @ B2 ) @ ( set_or4761464488706262899_set_a @ C @ D ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 )
| ( ( ord_le3724670747650509150_set_a @ C @ A2 )
& ( ord_le3724670747650509150_set_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_495_atLeastatMost__subset__iff,axiom,
! [A2: filter_a,B2: filter_a,C: filter_a,D: filter_a] :
( ( ord_le8471427362357671172lter_a @ ( set_or9026928291982918841lter_a @ A2 @ B2 ) @ ( set_or9026928291982918841lter_a @ C @ D ) )
= ( ~ ( ord_less_eq_filter_a @ A2 @ B2 )
| ( ( ord_less_eq_filter_a @ C @ A2 )
& ( ord_less_eq_filter_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_496_atLeastatMost__subset__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_497_atLeastatMost__subset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_498_psubsetD,axiom,
! [A: set_set_set_a,B: set_set_set_a,C: set_set_a] :
( ( ord_le8783759336481291722_set_a @ A @ B )
=> ( ( member_set_set_a @ C @ A )
=> ( member_set_set_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_499_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_500_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_501_psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_502_psubset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( ord_less_set_set_a @ B @ C2 )
=> ( ord_less_set_set_a @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_503_closed__atLeastAtMost,axiom,
! [A2: a,B2: a] : ( topolo784654279908865136osed_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) ) ).
% closed_atLeastAtMost
thf(fact_504_atLeastatMost__psubset__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a,D: set_set_a] :
( ( ord_le8783759336481291722_set_a @ ( set_or4761464488706262899_set_a @ A2 @ B2 ) @ ( set_or4761464488706262899_set_a @ C @ D ) )
= ( ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 )
| ( ( ord_le3724670747650509150_set_a @ C @ A2 )
& ( ord_le3724670747650509150_set_a @ B2 @ D )
& ( ( ord_less_set_set_a @ C @ A2 )
| ( ord_less_set_set_a @ B2 @ D ) ) ) )
& ( ord_le3724670747650509150_set_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_505_atLeastatMost__psubset__iff,axiom,
! [A2: filter_a,B2: filter_a,C: filter_a,D: filter_a] :
( ( ord_le2924155689415861776lter_a @ ( set_or9026928291982918841lter_a @ A2 @ B2 ) @ ( set_or9026928291982918841lter_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_filter_a @ A2 @ B2 )
| ( ( ord_less_eq_filter_a @ C @ A2 )
& ( ord_less_eq_filter_a @ B2 @ D )
& ( ( ord_less_filter_a @ C @ A2 )
| ( ord_less_filter_a @ B2 @ D ) ) ) )
& ( ord_less_eq_filter_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_506_atLeastatMost__psubset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D )
& ( ( ord_less_a @ C @ A2 )
| ( ord_less_a @ B2 @ D ) ) ) )
& ( ord_less_eq_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_507_atLeastatMost__psubset__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ord_less_set_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ B2 @ D )
& ( ( ord_less_set_a @ C @ A2 )
| ( ord_less_set_a @ B2 @ D ) ) ) )
& ( ord_less_eq_set_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_508_sup__lexord,axiom,
! [K: set_a > set_a,A: set_a,B: set_a,P: set_a > $o,C: set_a,S3: set_a] :
( ( ( ord_less_set_a @ ( K @ A ) @ ( K @ B ) )
=> ( P @ B ) )
=> ( ( ( ord_less_set_a @ ( K @ B ) @ ( K @ A ) )
=> ( P @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( P @ C ) )
=> ( ( ~ ( ord_less_eq_set_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_set_a @ ( K @ A ) @ ( K @ B ) )
=> ( P @ S3 ) ) )
=> ( P @ ( measur758946168800011932_set_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% sup_lexord
thf(fact_509_sup__lexord,axiom,
! [K: set_a > a,A: set_a,B: set_a,P: set_a > $o,C: set_a,S3: set_a] :
( ( ( ord_less_a @ ( K @ A ) @ ( K @ B ) )
=> ( P @ B ) )
=> ( ( ( ord_less_a @ ( K @ B ) @ ( K @ A ) )
=> ( P @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( P @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A ) @ ( K @ B ) )
=> ( P @ S3 ) ) )
=> ( P @ ( measur1530876966822229308et_a_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% sup_lexord
thf(fact_510_le__sup__lexord,axiom,
! [K: set_a > a,A: set_a,B: set_a,Ca: set_a,C: set_a,S3: set_a] :
( ( ( ord_less_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_set_a @ Ca @ B ) )
=> ( ( ( ord_less_a @ ( K @ B ) @ ( K @ A ) )
=> ( ord_less_eq_set_a @ Ca @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( ord_less_eq_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_set_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_set_a @ Ca @ ( measur1530876966822229308et_a_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_511_le__sup__lexord,axiom,
! [K: set_a > set_a,A: set_a,B: set_a,Ca: set_a,C: set_a,S3: set_a] :
( ( ( ord_less_set_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_set_a @ Ca @ B ) )
=> ( ( ( ord_less_set_a @ ( K @ B ) @ ( K @ A ) )
=> ( ord_less_eq_set_a @ Ca @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( ord_less_eq_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_set_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_set_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_set_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_set_a @ Ca @ ( measur758946168800011932_set_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_512_le__sup__lexord,axiom,
! [K: a > a,A: a,B: a,Ca: a,C: a,S3: a] :
( ( ( ord_less_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ B ) )
=> ( ( ( ord_less_a @ ( K @ B ) @ ( K @ A ) )
=> ( ord_less_eq_a @ Ca @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur6235662355876231836rd_a_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_513_le__sup__lexord,axiom,
! [K: a > set_a,A: a,B: a,Ca: a,C: a,S3: a] :
( ( ( ord_less_set_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ B ) )
=> ( ( ( ord_less_set_a @ ( K @ B ) @ ( K @ A ) )
=> ( ord_less_eq_a @ Ca @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_set_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_set_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur1314364908362886140_set_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_514_le__sup__lexord,axiom,
! [K: a > filter_a,A: a,B: a,Ca: a,C: a,S3: a] :
( ( ( ord_less_filter_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ B ) )
=> ( ( ( ord_less_filter_a @ ( K @ B ) @ ( K @ A ) )
=> ( ord_less_eq_a @ Ca @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_filter_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_filter_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur2310882534675631394lter_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_515_le__sup__lexord,axiom,
! [K: filter_a > a,A: filter_a,B: filter_a,Ca: filter_a,C: filter_a,S3: filter_a] :
( ( ( ord_less_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_filter_a @ Ca @ B ) )
=> ( ( ( ord_less_a @ ( K @ B ) @ ( K @ A ) )
=> ( ord_less_eq_filter_a @ Ca @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( ord_less_eq_filter_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_filter_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_filter_a @ Ca @ ( measur6254484919884501270er_a_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_516_le__sup__lexord,axiom,
! [K: set_a > filter_a,A: set_a,B: set_a,Ca: set_a,C: set_a,S3: set_a] :
( ( ( ord_less_filter_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_set_a @ Ca @ B ) )
=> ( ( ( ord_less_filter_a @ ( K @ B ) @ ( K @ A ) )
=> ( ord_less_eq_set_a @ Ca @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( ord_less_eq_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_filter_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_filter_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_set_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_set_a @ Ca @ ( measur9011977557813188546lter_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_517_le__sup__lexord,axiom,
! [K: a > set_set_a,A: a,B: a,Ca: a,C: a,S3: a] :
( ( ( ord_less_set_set_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ B ) )
=> ( ( ( ord_less_set_set_a @ ( K @ B ) @ ( K @ A ) )
=> ( ord_less_eq_a @ Ca @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ C ) )
=> ( ( ~ ( ord_le3724670747650509150_set_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_le3724670747650509150_set_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_a @ Ca @ ( measur3779986862995642716_set_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_518_le__sup__lexord,axiom,
! [K: set_set_a > a,A: set_set_a,B: set_set_a,Ca: set_set_a,C: set_set_a,S3: set_set_a] :
( ( ( ord_less_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_le3724670747650509150_set_a @ Ca @ B ) )
=> ( ( ( ord_less_a @ ( K @ B ) @ ( K @ A ) )
=> ( ord_le3724670747650509150_set_a @ Ca @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( ord_le3724670747650509150_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_le3724670747650509150_set_a @ Ca @ S3 ) ) )
=> ( ord_le3724670747650509150_set_a @ Ca @ ( measur7285218718371296220et_a_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_519_le__sup__lexord,axiom,
! [K: filter_a > set_a,A: filter_a,B: filter_a,Ca: filter_a,C: filter_a,S3: filter_a] :
( ( ( ord_less_set_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_filter_a @ Ca @ B ) )
=> ( ( ( ord_less_set_a @ ( K @ B ) @ ( K @ A ) )
=> ( ord_less_eq_filter_a @ Ca @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( ord_less_eq_filter_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_set_a @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_set_a @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_filter_a @ Ca @ S3 ) ) )
=> ( ord_less_eq_filter_a @ Ca @ ( measur5745845504613816950_set_a @ A @ B @ K @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_520_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a,D: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ A2 @ B2 ) @ ( set_or7717641506629902671_set_a @ C @ D ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ A2 )
& ( ord_less_set_set_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_521_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: filter_a,B2: filter_a,C: filter_a,D: filter_a] :
( ( ord_le8471427362357671172lter_a @ ( set_or9026928291982918841lter_a @ A2 @ B2 ) @ ( set_or8494873714499830933lter_a @ C @ D ) )
= ( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( ord_less_eq_filter_a @ C @ A2 )
& ( ord_less_filter_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_522_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or2348907005316661231_set_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_set_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_523_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or5139330845457685135Than_a @ C @ D ) )
= ( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_524_Icc__subset__Iic__iff,axiom,
! [L: set_set_a,H: set_set_a,H2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ L @ H ) @ ( set_or4016371710855203973_set_a @ H2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ L @ H )
| ( ord_le3724670747650509150_set_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_525_Icc__subset__Iic__iff,axiom,
! [L: filter_a,H: filter_a,H2: filter_a] :
( ( ord_le8471427362357671172lter_a @ ( set_or9026928291982918841lter_a @ L @ H ) @ ( set_or3848680440086517451lter_a @ H2 ) )
= ( ~ ( ord_less_eq_filter_a @ L @ H )
| ( ord_less_eq_filter_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_526_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_527_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_528_chain__subset__def,axiom,
( chain_subset_set_a
= ( ^ [C3: set_set_set_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ C3 )
=> ! [Y2: set_set_a] :
( ( member_set_set_a @ Y2 @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
| ( ord_le3724670747650509150_set_a @ Y2 @ X3 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_529_chain__subset__def,axiom,
( chain_subset_a
= ( ^ [C3: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ C3 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ C3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
| ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_530_atMost__eq__iff,axiom,
! [X2: a,Y3: a] :
( ( ( set_ord_atMost_a @ X2 )
= ( set_ord_atMost_a @ Y3 ) )
= ( X2 = Y3 ) ) ).
% atMost_eq_iff
thf(fact_531_atMost__eq__iff,axiom,
! [X2: set_a,Y3: set_a] :
( ( ( set_ord_atMost_set_a @ X2 )
= ( set_ord_atMost_set_a @ Y3 ) )
= ( X2 = Y3 ) ) ).
% atMost_eq_iff
thf(fact_532_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_533_atMost__iff,axiom,
! [I2: filter_a,K: filter_a] :
( ( member_filter_a @ I2 @ ( set_or3848680440086517451lter_a @ K ) )
= ( ord_less_eq_filter_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_534_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_535_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_536_greaterThanLessThan__iff,axiom,
! [I2: set_set_a,L: set_set_a,U3: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or5544419673604628330_set_a @ L @ U3 ) )
= ( ( ord_less_set_set_a @ L @ I2 )
& ( ord_less_set_set_a @ I2 @ U3 ) ) ) ).
% greaterThanLessThan_iff
thf(fact_537_greaterThanLessThan__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or6017932776736107018_set_a @ L @ U3 ) )
= ( ( ord_less_set_a @ L @ I2 )
& ( ord_less_set_a @ I2 @ U3 ) ) ) ).
% greaterThanLessThan_iff
thf(fact_538_greaterThanLessThan__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or5939364468397584554Than_a @ L @ U3 ) )
= ( ( ord_less_a @ L @ I2 )
& ( ord_less_a @ I2 @ U3 ) ) ) ).
% greaterThanLessThan_iff
thf(fact_539_atLeastLessThan__iff,axiom,
! [I2: set_set_a,L: set_set_a,U3: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or7717641506629902671_set_a @ L @ U3 ) )
= ( ( ord_le3724670747650509150_set_a @ L @ I2 )
& ( ord_less_set_set_a @ I2 @ U3 ) ) ) ).
% atLeastLessThan_iff
thf(fact_540_atLeastLessThan__iff,axiom,
! [I2: filter_a,L: filter_a,U3: filter_a] :
( ( member_filter_a @ I2 @ ( set_or8494873714499830933lter_a @ L @ U3 ) )
= ( ( ord_less_eq_filter_a @ L @ I2 )
& ( ord_less_filter_a @ I2 @ U3 ) ) ) ).
% atLeastLessThan_iff
thf(fact_541_atLeastLessThan__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or5139330845457685135Than_a @ L @ U3 ) )
= ( ( ord_less_eq_a @ L @ I2 )
& ( ord_less_a @ I2 @ U3 ) ) ) ).
% atLeastLessThan_iff
thf(fact_542_atLeastLessThan__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or2348907005316661231_set_a @ L @ U3 ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_set_a @ I2 @ U3 ) ) ) ).
% atLeastLessThan_iff
thf(fact_543_ivl__subset,axiom,
! [I2: a,J: a,M3: a,N: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ I2 @ J ) @ ( set_or5139330845457685135Than_a @ M3 @ N ) )
= ( ( ord_less_eq_a @ J @ I2 )
| ( ( ord_less_eq_a @ M3 @ I2 )
& ( ord_less_eq_a @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_544_atMost__subset__iff,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4016371710855203973_set_a @ X2 ) @ ( set_or4016371710855203973_set_a @ Y3 ) )
= ( ord_le3724670747650509150_set_a @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_545_atMost__subset__iff,axiom,
! [X2: filter_a,Y3: filter_a] :
( ( ord_le8471427362357671172lter_a @ ( set_or3848680440086517451lter_a @ X2 ) @ ( set_or3848680440086517451lter_a @ Y3 ) )
= ( ord_less_eq_filter_a @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_546_atMost__subset__iff,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X2 ) @ ( set_ord_atMost_set_a @ Y3 ) )
= ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_547_atMost__subset__iff,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ X2 ) @ ( set_ord_atMost_a @ Y3 ) )
= ( ord_less_eq_a @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_548_greaterThanAtMost__iff,axiom,
! [I2: set_set_a,L: set_set_a,U3: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or6073455820589966094_set_a @ L @ U3 ) )
= ( ( ord_less_set_set_a @ L @ I2 )
& ( ord_le3724670747650509150_set_a @ I2 @ U3 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_549_greaterThanAtMost__iff,axiom,
! [I2: filter_a,L: filter_a,U3: filter_a] :
( ( member_filter_a @ I2 @ ( set_or2727982546503234516lter_a @ L @ U3 ) )
= ( ( ord_less_filter_a @ L @ I2 )
& ( ord_less_eq_filter_a @ I2 @ U3 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_550_greaterThanAtMost__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or4472690218693186638Most_a @ L @ U3 ) )
= ( ( ord_less_a @ L @ I2 )
& ( ord_less_eq_a @ I2 @ U3 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_551_greaterThanAtMost__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or2503527069484367278_set_a @ L @ U3 ) )
= ( ( ord_less_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U3 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_552_sup__lexord1,axiom,
! [K: set_a > set_a,A: set_a,B: set_a,S3: set_a,C: set_a] :
( ( ( K @ A )
= ( K @ B ) )
=> ( ( measur758946168800011932_set_a @ A @ B @ K @ S3 @ C )
= C ) ) ).
% sup_lexord1
thf(fact_553_sup__lexord1,axiom,
! [K: set_a > a,A: set_a,B: set_a,S3: set_a,C: set_a] :
( ( ( K @ A )
= ( K @ B ) )
=> ( ( measur1530876966822229308et_a_a @ A @ B @ K @ S3 @ C )
= C ) ) ).
% sup_lexord1
thf(fact_554_sup__lexord__commute,axiom,
( measur758946168800011932_set_a
= ( ^ [A5: set_a,B5: set_a] : ( measur758946168800011932_set_a @ B5 @ A5 ) ) ) ).
% sup_lexord_commute
thf(fact_555_sup__lexord__commute,axiom,
( measur1530876966822229308et_a_a
= ( ^ [A5: set_a,B5: set_a] : ( measur1530876966822229308et_a_a @ B5 @ A5 ) ) ) ).
% sup_lexord_commute
thf(fact_556_atLeastLessThan__eq__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ C @ D )
=> ( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D ) )
= ( ( A2 = C )
& ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_557_Ico__eq__Ico,axiom,
! [L: a,H: a,L2: a,H2: a] :
( ( ( set_or5139330845457685135Than_a @ L @ H )
= ( set_or5139330845457685135Than_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_a @ L @ H )
& ~ ( ord_less_a @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_558_atLeastLessThan__inj_I1_J,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ C @ D )
=> ( A2 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_559_atLeastLessThan__inj_I2_J,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ C @ D )
=> ( B2 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_560_Ioc__inj,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( set_or4472690218693186638Most_a @ A2 @ B2 )
= ( set_or4472690218693186638Most_a @ C @ D ) )
= ( ( ( ord_less_eq_a @ B2 @ A2 )
& ( ord_less_eq_a @ D @ C ) )
| ( ( A2 = C )
& ( B2 = D ) ) ) ) ).
% Ioc_inj
thf(fact_561_closed__atMost,axiom,
! [A2: a] : ( topolo784654279908865136osed_a @ ( set_ord_atMost_a @ A2 ) ) ).
% closed_atMost
thf(fact_562_open__greaterThanLessThan,axiom,
! [A2: a,B2: a] : ( topolo8477419352202985285open_a @ ( set_or5939364468397584554Than_a @ A2 @ B2 ) ) ).
% open_greaterThanLessThan
thf(fact_563_atLeastLessThan__subset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ A2 @ B2 ) @ ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_eq_a @ B2 @ A2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_564_Ioc__subset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ A2 @ B2 ) @ ( set_or4472690218693186638Most_a @ C @ D ) )
= ( ( ord_less_eq_a @ B2 @ A2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_565_sup__lexord__def,axiom,
( measur758946168800011932_set_a
= ( ^ [A5: set_a,B5: set_a,K2: set_a > set_a,S4: set_a,C4: set_a] :
( if_set_a
@ ( ( K2 @ A5 )
= ( K2 @ B5 ) )
@ C4
@ ( if_set_a
@ ( ~ ( ord_less_eq_set_a @ ( K2 @ A5 ) @ ( K2 @ B5 ) )
& ~ ( ord_less_eq_set_a @ ( K2 @ B5 ) @ ( K2 @ A5 ) ) )
@ S4
@ ( if_set_a @ ( ord_less_eq_set_a @ ( K2 @ B5 ) @ ( K2 @ A5 ) ) @ A5 @ B5 ) ) ) ) ) ).
% sup_lexord_def
thf(fact_566_sup__lexord__def,axiom,
( measur1530876966822229308et_a_a
= ( ^ [A5: set_a,B5: set_a,K2: set_a > a,S4: set_a,C4: set_a] :
( if_set_a
@ ( ( K2 @ A5 )
= ( K2 @ B5 ) )
@ C4
@ ( if_set_a
@ ( ~ ( ord_less_eq_a @ ( K2 @ A5 ) @ ( K2 @ B5 ) )
& ~ ( ord_less_eq_a @ ( K2 @ B5 ) @ ( K2 @ A5 ) ) )
@ S4
@ ( if_set_a @ ( ord_less_eq_a @ ( K2 @ B5 ) @ ( K2 @ A5 ) ) @ A5 @ B5 ) ) ) ) ) ).
% sup_lexord_def
thf(fact_567_open__right,axiom,
! [S: set_a,X2: a,Y3: a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( member_a @ X2 @ S )
=> ( ( ord_less_a @ X2 @ Y3 )
=> ? [B3: a] :
( ( ord_less_a @ X2 @ B3 )
& ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ X2 @ B3 ) @ S ) ) ) ) ) ).
% open_right
thf(fact_568_open__left,axiom,
! [S: set_a,X2: a,Y3: a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( member_a @ X2 @ S )
=> ( ( ord_less_a @ Y3 @ X2 )
=> ? [B3: a] :
( ( ord_less_a @ B3 @ X2 )
& ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ B3 @ X2 ) @ S ) ) ) ) ) ).
% open_left
thf(fact_569_ivl__disj__un__two_I5_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_a @ L @ M3 )
=> ( ( ord_less_eq_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M3 ) @ ( set_or672772299803893939Most_a @ M3 @ U3 ) )
= ( set_or4472690218693186638Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_570_ivl__disj__un__two_I4_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_eq_a @ L @ M3 )
=> ( ( ord_less_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M3 ) @ ( set_or5939364468397584554Than_a @ M3 @ U3 ) )
= ( set_or5139330845457685135Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_571_ivl__disj__un__two__touch_I1_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_a @ L @ M3 )
=> ( ( ord_less_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M3 ) @ ( set_or5139330845457685135Than_a @ M3 @ U3 ) )
= ( set_or5939364468397584554Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_572_ivl__disj__un__two_I2_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_eq_a @ L @ M3 )
=> ( ( ord_less_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M3 ) @ ( set_or5939364468397584554Than_a @ M3 @ U3 ) )
= ( set_or5939364468397584554Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_573_ivl__disj__un__two_I1_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_a @ L @ M3 )
=> ( ( ord_less_eq_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M3 ) @ ( set_or5139330845457685135Than_a @ M3 @ U3 ) )
= ( set_or5939364468397584554Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_574_ivl__disj__un__two__touch_I3_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_a @ L @ M3 )
=> ( ( ord_less_eq_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M3 ) @ ( set_or672772299803893939Most_a @ M3 @ U3 ) )
= ( set_or4472690218693186638Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_575_ivl__disj__un__two__touch_I2_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_eq_a @ L @ M3 )
=> ( ( ord_less_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M3 ) @ ( set_or5139330845457685135Than_a @ M3 @ U3 ) )
= ( set_or5139330845457685135Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_576_atLeastAtMost__def,axiom,
( set_or6288561110385358355_set_a
= ( ^ [L3: set_a,U4: set_a] : ( inf_inf_set_set_a @ ( set_or8362275514725411625_set_a @ L3 ) @ ( set_ord_atMost_set_a @ U4 ) ) ) ) ).
% atLeastAtMost_def
thf(fact_577_atLeastAtMost__def,axiom,
( set_or672772299803893939Most_a
= ( ^ [L3: a,U4: a] : ( inf_inf_set_a @ ( set_ord_atLeast_a @ L3 ) @ ( set_ord_atMost_a @ U4 ) ) ) ) ).
% atLeastAtMost_def
thf(fact_578_Int__iff,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) )
= ( ( member_set_set_a @ C @ A )
& ( member_set_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_579_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_580_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_581_IntI,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 @ B )
=> ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_582_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_583_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_584_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_585_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_586_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_587_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_588_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_589_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_590_Int__subset__iff,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A )
& ( ord_le3724670747650509150_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_591_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_592_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_593_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_594_open__Int,axiom,
! [S: set_a,T5: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( topolo8477419352202985285open_a @ T5 )
=> ( topolo8477419352202985285open_a @ ( inf_inf_set_a @ S @ T5 ) ) ) ) ).
% open_Int
thf(fact_595_Un__Int__eq_I1_J,axiom,
! [S: set_a,T5: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T5 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_596_Un__Int__eq_I1_J,axiom,
! [S: set_set_a,T5: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T5 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_597_Un__Int__eq_I2_J,axiom,
! [S: set_a,T5: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T5 ) @ T5 )
= T5 ) ).
% Un_Int_eq(2)
thf(fact_598_Un__Int__eq_I2_J,axiom,
! [S: set_set_a,T5: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T5 ) @ T5 )
= T5 ) ).
% Un_Int_eq(2)
thf(fact_599_Un__Int__eq_I3_J,axiom,
! [S: set_a,T5: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T5 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_600_Un__Int__eq_I3_J,axiom,
! [S: set_set_a,T5: set_set_a] :
( ( inf_inf_set_set_a @ S @ ( sup_sup_set_set_a @ S @ T5 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_601_Un__Int__eq_I4_J,axiom,
! [T5: set_a,S: set_a] :
( ( inf_inf_set_a @ T5 @ ( sup_sup_set_a @ S @ T5 ) )
= T5 ) ).
% Un_Int_eq(4)
thf(fact_602_Un__Int__eq_I4_J,axiom,
! [T5: set_set_a,S: set_set_a] :
( ( inf_inf_set_set_a @ T5 @ ( sup_sup_set_set_a @ S @ T5 ) )
= T5 ) ).
% Un_Int_eq(4)
thf(fact_603_Int__Un__eq_I1_J,axiom,
! [S: set_a,T5: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T5 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_604_Int__Un__eq_I1_J,axiom,
! [S: set_set_a,T5: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T5 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_605_Int__Un__eq_I2_J,axiom,
! [S: set_a,T5: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T5 ) @ T5 )
= T5 ) ).
% Int_Un_eq(2)
thf(fact_606_Int__Un__eq_I2_J,axiom,
! [S: set_set_a,T5: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T5 ) @ T5 )
= T5 ) ).
% Int_Un_eq(2)
thf(fact_607_Int__Un__eq_I3_J,axiom,
! [S: set_a,T5: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T5 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_608_Int__Un__eq_I3_J,axiom,
! [S: set_set_a,T5: set_set_a] :
( ( sup_sup_set_set_a @ S @ ( inf_inf_set_set_a @ S @ T5 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_609_Int__Un__eq_I4_J,axiom,
! [T5: set_a,S: set_a] :
( ( sup_sup_set_a @ T5 @ ( inf_inf_set_a @ S @ T5 ) )
= T5 ) ).
% Int_Un_eq(4)
thf(fact_610_Int__Un__eq_I4_J,axiom,
! [T5: set_set_a,S: set_set_a] :
( ( sup_sup_set_set_a @ T5 @ ( inf_inf_set_set_a @ S @ T5 ) )
= T5 ) ).
% Int_Un_eq(4)
thf(fact_611_open__Un,axiom,
! [S: set_a,T5: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( topolo8477419352202985285open_a @ T5 )
=> ( topolo8477419352202985285open_a @ ( sup_sup_set_a @ S @ T5 ) ) ) ) ).
% open_Un
thf(fact_612_closed__Int,axiom,
! [S: set_a,T5: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo784654279908865136osed_a @ T5 )
=> ( topolo784654279908865136osed_a @ ( inf_inf_set_a @ S @ T5 ) ) ) ) ).
% closed_Int
thf(fact_613_closed__Un,axiom,
! [S: set_a,T5: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo784654279908865136osed_a @ T5 )
=> ( topolo784654279908865136osed_a @ ( sup_sup_set_a @ S @ T5 ) ) ) ) ).
% closed_Un
thf(fact_614_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_615_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_616_Int__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_617_Int__left__commute,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) )
= ( inf_inf_set_set_a @ B @ ( inf_inf_set_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_618_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_619_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_620_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_621_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_622_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_623_Int__left__absorb,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_624_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_625_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_626_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_627_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_628_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_629_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_630_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_631_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_632_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_633_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_634_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_635_Int__commute,axiom,
( inf_inf_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] : ( inf_inf_set_set_a @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_636_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_637_Un__commute,axiom,
( sup_sup_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] : ( sup_sup_set_set_a @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_638_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_639_Int__absorb,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_640_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_641_Un__absorb,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_642_Int__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_643_Int__assoc,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_644_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_645_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_646_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_647_ball__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( sup_sup_set_set_a @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_648_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_649_bex__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( sup_sup_set_set_a @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: set_a] :
( ( member_set_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_650_IntD2,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) )
=> ( member_set_set_a @ C @ B ) ) ).
% IntD2
thf(fact_651_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_652_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_653_IntD1,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) )
=> ( member_set_set_a @ C @ A ) ) ).
% IntD1
thf(fact_654_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_655_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_656_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_657_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_658_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_659_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_660_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_661_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_662_IntE,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) )
=> ~ ( ( member_set_set_a @ C @ A )
=> ~ ( member_set_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_663_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_664_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_665_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_666_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_667_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_668_subset__Un__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( sup_sup_set_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_669_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_670_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 ) )
=> ~ ! [A7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ A )
=> ! [B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B7 @ B )
=> ( C2
!= ( sup_sup_set_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_671_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B )
=> ( C2
!= ( sup_sup_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_672_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_673_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_674_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_675_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_676_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_677_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_678_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_679_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_680_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_681_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_682_Un__mono,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_683_Un__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_684_Int__Collect__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a,P: set_set_a > $o,Q: set_set_a > $o] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ! [X: set_set_a] :
( ( member_set_set_a @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le5722252365846178494_set_a @ ( inf_in1205276777018777868_set_a @ A @ ( collect_set_set_a @ P ) ) @ ( inf_in1205276777018777868_set_a @ B @ ( collect_set_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_685_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 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( 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_686_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( 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_687_Int__greatest,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_688_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_689_Int__absorb2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( inf_inf_set_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_690_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_691_Int__absorb1,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( inf_inf_set_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_692_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_693_Int__lower2,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_694_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_695_Int__lower1,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_696_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_697_Int__mono,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_698_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_699_ivl__disj__un__two__touch_I4_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_eq_a @ L @ M3 )
=> ( ( ord_less_eq_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M3 ) @ ( set_or672772299803893939Most_a @ M3 @ U3 ) )
= ( set_or672772299803893939Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_700_ivl__disj__un__two_I3_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_eq_a @ L @ M3 )
=> ( ( ord_less_eq_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ M3 ) @ ( set_or5139330845457685135Than_a @ M3 @ U3 ) )
= ( set_or5139330845457685135Than_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_701_ivl__disj__un__two_I6_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_eq_a @ L @ M3 )
=> ( ( ord_less_eq_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M3 ) @ ( set_or4472690218693186638Most_a @ M3 @ U3 ) )
= ( set_or4472690218693186638Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_702_ivl__disj__un__two_I7_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_eq_a @ L @ M3 )
=> ( ( ord_less_eq_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ M3 ) @ ( set_or672772299803893939Most_a @ M3 @ U3 ) )
= ( set_or672772299803893939Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_703_ivl__disj__un__two_I8_J,axiom,
! [L: a,M3: a,U3: a] :
( ( ord_less_eq_a @ L @ M3 )
=> ( ( ord_less_eq_a @ M3 @ U3 )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M3 ) @ ( set_or4472690218693186638Most_a @ M3 @ U3 ) )
= ( set_or672772299803893939Most_a @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_704_ivl__disj__un__one_I8_J,axiom,
! [L: a,U3: a] :
( ( ord_less_eq_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ U3 ) @ ( set_ord_atLeast_a @ U3 ) )
= ( set_ord_atLeast_a @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_705_ivl__disj__un__one_I3_J,axiom,
! [L: a,U3: a] :
( ( ord_less_eq_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or4472690218693186638Most_a @ L @ U3 ) )
= ( set_ord_atMost_a @ U3 ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_706_inf__sup__absorb,axiom,
! [X2: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_707_inf__sup__absorb,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y3 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_708_sup__inf__absorb,axiom,
! [X2: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y3 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_709_sup__inf__absorb,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ X2 @ Y3 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_710_le__sup__iff,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X2 @ Y3 ) @ Z2 )
= ( ( ord_le3724670747650509150_set_a @ X2 @ Z2 )
& ( ord_le3724670747650509150_set_a @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_711_le__sup__iff,axiom,
! [X2: filter_a,Y3: filter_a,Z2: filter_a] :
( ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ X2 @ Y3 ) @ Z2 )
= ( ( ord_less_eq_filter_a @ X2 @ Z2 )
& ( ord_less_eq_filter_a @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_712_le__sup__iff,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ Z2 )
= ( ( ord_less_eq_set_a @ X2 @ Z2 )
& ( ord_less_eq_set_a @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_713_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_714_sup_Obounded__iff,axiom,
! [B2: filter_a,C: filter_a,A2: filter_a] :
( ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_filter_a @ B2 @ A2 )
& ( ord_less_eq_filter_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_715_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_716_le__inf__iff,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ ( inf_inf_set_set_a @ Y3 @ Z2 ) )
= ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
& ( ord_le3724670747650509150_set_a @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_717_le__inf__iff,axiom,
! [X2: filter_a,Y3: filter_a,Z2: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ ( inf_inf_filter_a @ Y3 @ Z2 ) )
= ( ( ord_less_eq_filter_a @ X2 @ Y3 )
& ( ord_less_eq_filter_a @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_718_le__inf__iff,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( ord_less_eq_set_a @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_719_inf_Obounded__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_720_inf_Obounded__iff,axiom,
! [A2: filter_a,B2: filter_a,C: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ ( inf_inf_filter_a @ B2 @ C ) )
= ( ( ord_less_eq_filter_a @ A2 @ B2 )
& ( ord_less_eq_filter_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_721_inf_Obounded__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_722_distrib__inf__le,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ ( inf_inf_set_set_a @ X2 @ Z2 ) ) @ ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_723_distrib__inf__le,axiom,
! [X2: filter_a,Y3: filter_a,Z2: filter_a] : ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ ( inf_inf_filter_a @ X2 @ Y3 ) @ ( inf_inf_filter_a @ X2 @ Z2 ) ) @ ( inf_inf_filter_a @ X2 @ ( sup_sup_filter_a @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_724_distrib__inf__le,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z2 ) ) @ ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_725_inf__right__idem,axiom,
! [X2: set_a,Y3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ Y3 )
= ( inf_inf_set_a @ X2 @ Y3 ) ) ).
% inf_right_idem
thf(fact_726_inf__right__idem,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ Y3 )
= ( inf_inf_set_set_a @ X2 @ Y3 ) ) ).
% inf_right_idem
thf(fact_727_inf_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_728_inf_Oright__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_729_inf__left__idem,axiom,
! [X2: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y3 ) )
= ( inf_inf_set_a @ X2 @ Y3 ) ) ).
% inf_left_idem
thf(fact_730_inf__left__idem,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( inf_inf_set_set_a @ X2 @ Y3 ) )
= ( inf_inf_set_set_a @ X2 @ Y3 ) ) ).
% inf_left_idem
thf(fact_731_inf_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_732_inf_Oleft__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_733_inf__idem,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_734_inf__idem,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_735_inf_Oidem,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_736_inf_Oidem,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_737_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_738_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_739_sup__left__idem,axiom,
! [X2: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) )
= ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% sup_left_idem
thf(fact_740_sup__left__idem,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y3 ) )
= ( sup_sup_set_set_a @ X2 @ Y3 ) ) ).
% sup_left_idem
thf(fact_741_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_742_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_743_sup__idem,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_744_sup__idem,axiom,
! [X2: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_745_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_746_sup_Oidem,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_747_inf__left__commute,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_748_inf__left__commute,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_set_a @ Y3 @ ( inf_inf_set_set_a @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_749_inf_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C ) )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_750_inf_Oleft__commute,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ B2 @ ( inf_inf_set_set_a @ A2 @ C ) )
= ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_751_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X3 ) ) ) ).
% inf_commute
thf(fact_752_inf__commute,axiom,
( inf_inf_set_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X3 ) ) ) ).
% inf_commute
thf(fact_753_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_754_inf_Ocommute,axiom,
( inf_inf_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] : ( inf_inf_set_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_755_inf__assoc,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ Z2 )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_756_inf__assoc,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ Z2 )
= ( inf_inf_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_757_inf_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_758_inf_Oassoc,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_759_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_760_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_761_inf__sup__aci_I2_J,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ Z2 )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_762_inf__sup__aci_I2_J,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ Z2 )
= ( inf_inf_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_763_inf__sup__aci_I3_J,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_764_inf__sup__aci_I3_J,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_set_a @ Y3 @ ( inf_inf_set_set_a @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_765_inf__sup__aci_I4_J,axiom,
! [X2: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y3 ) )
= ( inf_inf_set_a @ X2 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_766_inf__sup__aci_I4_J,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( inf_inf_set_set_a @ X2 @ Y3 ) )
= ( inf_inf_set_set_a @ X2 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_767_sup__left__commute,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_768_sup__left__commute,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_set_a @ Y3 @ ( sup_sup_set_set_a @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_769_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_770_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_771_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X3 ) ) ) ).
% sup_commute
thf(fact_772_sup__commute,axiom,
( sup_sup_set_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] : ( sup_sup_set_set_a @ Y2 @ X3 ) ) ) ).
% sup_commute
thf(fact_773_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_774_sup_Ocommute,axiom,
( sup_sup_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] : ( sup_sup_set_set_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_775_sup__assoc,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ Z2 )
= ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_776_sup__assoc,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y3 ) @ Z2 )
= ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y3 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_777_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_778_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_779_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_780_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] : ( sup_sup_set_set_a @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_781_inf__sup__aci_I6_J,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ Z2 )
= ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_782_inf__sup__aci_I6_J,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y3 ) @ Z2 )
= ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_783_inf__sup__aci_I7_J,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_784_inf__sup__aci_I7_J,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_set_a @ Y3 @ ( sup_sup_set_set_a @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_785_inf__sup__aci_I8_J,axiom,
! [X2: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) )
= ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_786_inf__sup__aci_I8_J,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y3 ) )
= ( sup_sup_set_set_a @ X2 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_787_inf_OcoboundedI2,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_788_inf_OcoboundedI2,axiom,
! [B2: filter_a,C: filter_a,A2: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ C )
=> ( ord_less_eq_filter_a @ ( inf_inf_filter_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_789_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_790_inf_OcoboundedI1,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_791_inf_OcoboundedI1,axiom,
! [A2: filter_a,C: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ C )
=> ( ord_less_eq_filter_a @ ( inf_inf_filter_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_792_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_793_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_794_inf_Oabsorb__iff2,axiom,
( ord_less_eq_filter_a
= ( ^ [B4: filter_a,A4: filter_a] :
( ( inf_inf_filter_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_795_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_796_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_797_inf_Oabsorb__iff1,axiom,
( ord_less_eq_filter_a
= ( ^ [A4: filter_a,B4: filter_a] :
( ( inf_inf_filter_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_798_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_799_inf_Ocobounded2,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_800_inf_Ocobounded2,axiom,
! [A2: filter_a,B2: filter_a] : ( ord_less_eq_filter_a @ ( inf_inf_filter_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_801_inf_Ocobounded2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_802_inf_Ocobounded1,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_803_inf_Ocobounded1,axiom,
! [A2: filter_a,B2: filter_a] : ( ord_less_eq_filter_a @ ( inf_inf_filter_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_804_inf_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_805_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( A4
= ( inf_inf_set_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_806_inf_Oorder__iff,axiom,
( ord_less_eq_filter_a
= ( ^ [A4: filter_a,B4: filter_a] :
( A4
= ( inf_inf_filter_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_807_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_808_inf__greatest,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( inf_inf_set_set_a @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_809_inf__greatest,axiom,
! [X2: filter_a,Y3: filter_a,Z2: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ Y3 )
=> ( ( ord_less_eq_filter_a @ X2 @ Z2 )
=> ( ord_less_eq_filter_a @ X2 @ ( inf_inf_filter_a @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_810_inf__greatest,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ X2 @ Z2 )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_811_inf_OboundedI,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_812_inf_OboundedI,axiom,
! [A2: filter_a,B2: filter_a,C: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( ord_less_eq_filter_a @ A2 @ C )
=> ( ord_less_eq_filter_a @ A2 @ ( inf_inf_filter_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_813_inf_OboundedI,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_814_inf_OboundedE,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_815_inf_OboundedE,axiom,
! [A2: filter_a,B2: filter_a,C: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ ( inf_inf_filter_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ~ ( ord_less_eq_filter_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_816_inf_OboundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_817_inf__absorb2,axiom,
! [Y3: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X2 )
=> ( ( inf_inf_set_set_a @ X2 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_818_inf__absorb2,axiom,
! [Y3: filter_a,X2: filter_a] :
( ( ord_less_eq_filter_a @ Y3 @ X2 )
=> ( ( inf_inf_filter_a @ X2 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_819_inf__absorb2,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( inf_inf_set_a @ X2 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_820_inf__absorb1,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( inf_inf_set_set_a @ X2 @ Y3 )
= X2 ) ) ).
% inf_absorb1
thf(fact_821_inf__absorb1,axiom,
! [X2: filter_a,Y3: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ Y3 )
=> ( ( inf_inf_filter_a @ X2 @ Y3 )
= X2 ) ) ).
% inf_absorb1
thf(fact_822_inf__absorb1,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( inf_inf_set_a @ X2 @ Y3 )
= X2 ) ) ).
% inf_absorb1
thf(fact_823_inf_Oabsorb2,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_824_inf_Oabsorb2,axiom,
! [B2: filter_a,A2: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ A2 )
=> ( ( inf_inf_filter_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_825_inf_Oabsorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_826_inf_Oabsorb1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_827_inf_Oabsorb1,axiom,
! [A2: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( inf_inf_filter_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_828_inf_Oabsorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_829_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_830_le__iff__inf,axiom,
( ord_less_eq_filter_a
= ( ^ [X3: filter_a,Y2: filter_a] :
( ( inf_inf_filter_a @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_831_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_832_inf__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X2: set_set_a,Y3: set_set_a] :
( ! [X: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: set_set_a,Y4: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y4 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Z4 )
=> ( ord_le3724670747650509150_set_a @ X @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_set_set_a @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_833_inf__unique,axiom,
! [F: filter_a > filter_a > filter_a,X2: filter_a,Y3: filter_a] :
( ! [X: filter_a,Y4: filter_a] : ( ord_less_eq_filter_a @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: filter_a,Y4: filter_a] : ( ord_less_eq_filter_a @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: filter_a,Y4: filter_a,Z4: filter_a] :
( ( ord_less_eq_filter_a @ X @ Y4 )
=> ( ( ord_less_eq_filter_a @ X @ Z4 )
=> ( ord_less_eq_filter_a @ X @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_filter_a @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_834_inf__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y3: set_a] :
( ! [X: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: set_a,Y4: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ( ord_less_eq_set_a @ X @ Z4 )
=> ( ord_less_eq_set_a @ X @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_835_inf_OorderI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2
= ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_836_inf_OorderI,axiom,
! [A2: filter_a,B2: filter_a] :
( ( A2
= ( inf_inf_filter_a @ A2 @ B2 ) )
=> ( ord_less_eq_filter_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_837_inf_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_838_inf_OorderE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_839_inf_OorderE,axiom,
! [A2: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_filter_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_840_inf_OorderE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_841_le__infI2,axiom,
! [B2: set_set_a,X2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_842_le__infI2,axiom,
! [B2: filter_a,X2: filter_a,A2: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ X2 )
=> ( ord_less_eq_filter_a @ ( inf_inf_filter_a @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_843_le__infI2,axiom,
! [B2: set_a,X2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_844_le__infI1,axiom,
! [A2: set_set_a,X2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_845_le__infI1,axiom,
! [A2: filter_a,X2: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ X2 )
=> ( ord_less_eq_filter_a @ ( inf_inf_filter_a @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_846_le__infI1,axiom,
! [A2: set_a,X2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_847_inf__mono,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_848_inf__mono,axiom,
! [A2: filter_a,C: filter_a,B2: filter_a,D: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ C )
=> ( ( ord_less_eq_filter_a @ B2 @ D )
=> ( ord_less_eq_filter_a @ ( inf_inf_filter_a @ A2 @ B2 ) @ ( inf_inf_filter_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_849_inf__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_850_le__infI,axiom,
! [X2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_851_le__infI,axiom,
! [X2: filter_a,A2: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ A2 )
=> ( ( ord_less_eq_filter_a @ X2 @ B2 )
=> ( ord_less_eq_filter_a @ X2 @ ( inf_inf_filter_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_852_le__infI,axiom,
! [X2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ B2 )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_853_le__infE,axiom,
! [X2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X2 @ A2 )
=> ~ ( ord_le3724670747650509150_set_a @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_854_le__infE,axiom,
! [X2: filter_a,A2: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ ( inf_inf_filter_a @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_filter_a @ X2 @ A2 )
=> ~ ( ord_less_eq_filter_a @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_855_le__infE,axiom,
! [X2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X2 @ A2 )
=> ~ ( ord_less_eq_set_a @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_856_inf__le2,axiom,
! [X2: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_857_inf__le2,axiom,
! [X2: filter_a,Y3: filter_a] : ( ord_less_eq_filter_a @ ( inf_inf_filter_a @ X2 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_858_inf__le2,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_859_inf__le1,axiom,
! [X2: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_860_inf__le1,axiom,
! [X2: filter_a,Y3: filter_a] : ( ord_less_eq_filter_a @ ( inf_inf_filter_a @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_861_inf__le1,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_862_inf__sup__ord_I1_J,axiom,
! [X2: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_863_inf__sup__ord_I1_J,axiom,
! [X2: filter_a,Y3: filter_a] : ( ord_less_eq_filter_a @ ( inf_inf_filter_a @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_864_inf__sup__ord_I1_J,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_865_inf__sup__ord_I2_J,axiom,
! [X2: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_866_inf__sup__ord_I2_J,axiom,
! [X2: filter_a,Y3: filter_a] : ( ord_less_eq_filter_a @ ( inf_inf_filter_a @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_867_inf__sup__ord_I2_J,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_868_inf_Ostrict__coboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_869_inf_Ostrict__coboundedI2,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_870_inf_Ostrict__coboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_871_inf_Ostrict__coboundedI1,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ C )
=> ( ord_less_set_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_872_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_873_inf_Ostrict__order__iff,axiom,
( ord_less_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( A4
= ( inf_inf_set_set_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_874_inf_Ostrict__boundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_875_inf_Ostrict__boundedE,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_set_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_876_inf_Oabsorb4,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_877_inf_Oabsorb4,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_878_inf_Oabsorb3,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_879_inf_Oabsorb3,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_880_less__infI2,axiom,
! [B2: set_a,X2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ X2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_881_less__infI2,axiom,
! [B2: set_set_a,X2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ X2 )
=> ( ord_less_set_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_882_less__infI1,axiom,
! [A2: set_a,X2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ X2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_883_less__infI1,axiom,
! [A2: set_set_a,X2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ X2 )
=> ( ord_less_set_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_884_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_885_sup_OcoboundedI2,axiom,
! [C: filter_a,B2: filter_a,A2: filter_a] :
( ( ord_less_eq_filter_a @ C @ B2 )
=> ( ord_less_eq_filter_a @ C @ ( sup_sup_filter_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_886_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_887_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_888_sup_OcoboundedI1,axiom,
! [C: filter_a,A2: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ C @ A2 )
=> ( ord_less_eq_filter_a @ C @ ( sup_sup_filter_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_889_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_890_sup_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_891_sup_Oabsorb__iff2,axiom,
( ord_less_eq_filter_a
= ( ^ [A4: filter_a,B4: filter_a] :
( ( sup_sup_filter_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_892_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_893_sup_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_894_sup_Oabsorb__iff1,axiom,
( ord_less_eq_filter_a
= ( ^ [B4: filter_a,A4: filter_a] :
( ( sup_sup_filter_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_895_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_896_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_897_sup_Ocobounded2,axiom,
! [B2: filter_a,A2: filter_a] : ( ord_less_eq_filter_a @ B2 @ ( sup_sup_filter_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_898_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_899_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_900_sup_Ocobounded1,axiom,
! [A2: filter_a,B2: filter_a] : ( ord_less_eq_filter_a @ A2 @ ( sup_sup_filter_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_901_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_902_sup_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( A4
= ( sup_sup_set_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_903_sup_Oorder__iff,axiom,
( ord_less_eq_filter_a
= ( ^ [B4: filter_a,A4: filter_a] :
( A4
= ( sup_sup_filter_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_904_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_905_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_906_sup_OboundedI,axiom,
! [B2: filter_a,A2: filter_a,C: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ A2 )
=> ( ( ord_less_eq_filter_a @ C @ A2 )
=> ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_907_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_908_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_909_sup_OboundedE,axiom,
! [B2: filter_a,C: filter_a,A2: filter_a] :
( ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_filter_a @ B2 @ A2 )
=> ~ ( ord_less_eq_filter_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_910_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_911_sup__absorb2,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( sup_sup_set_set_a @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_912_sup__absorb2,axiom,
! [X2: filter_a,Y3: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ Y3 )
=> ( ( sup_sup_filter_a @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_913_sup__absorb2,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( sup_sup_set_a @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_914_sup__absorb1,axiom,
! [Y3: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X2 )
=> ( ( sup_sup_set_set_a @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_915_sup__absorb1,axiom,
! [Y3: filter_a,X2: filter_a] :
( ( ord_less_eq_filter_a @ Y3 @ X2 )
=> ( ( sup_sup_filter_a @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_916_sup__absorb1,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( sup_sup_set_a @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_917_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_918_sup_Oabsorb2,axiom,
! [A2: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ B2 )
=> ( ( sup_sup_filter_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_919_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_920_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_921_sup_Oabsorb1,axiom,
! [B2: filter_a,A2: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ A2 )
=> ( ( sup_sup_filter_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_922_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_923_sup__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X2: set_set_a,Y3: set_set_a] :
( ! [X: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ ( F @ X @ Y4 ) )
=> ( ! [X: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ Y4 @ ( F @ X @ Y4 ) )
=> ( ! [X: set_set_a,Y4: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y4 @ X )
=> ( ( ord_le3724670747650509150_set_a @ Z4 @ X )
=> ( ord_le3724670747650509150_set_a @ ( F @ Y4 @ Z4 ) @ X ) ) )
=> ( ( sup_sup_set_set_a @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_924_sup__unique,axiom,
! [F: filter_a > filter_a > filter_a,X2: filter_a,Y3: filter_a] :
( ! [X: filter_a,Y4: filter_a] : ( ord_less_eq_filter_a @ X @ ( F @ X @ Y4 ) )
=> ( ! [X: filter_a,Y4: filter_a] : ( ord_less_eq_filter_a @ Y4 @ ( F @ X @ Y4 ) )
=> ( ! [X: filter_a,Y4: filter_a,Z4: filter_a] :
( ( ord_less_eq_filter_a @ Y4 @ X )
=> ( ( ord_less_eq_filter_a @ Z4 @ X )
=> ( ord_less_eq_filter_a @ ( F @ Y4 @ Z4 ) @ X ) ) )
=> ( ( sup_sup_filter_a @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_925_sup__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y3: set_a] :
( ! [X: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X @ ( F @ X @ Y4 ) )
=> ( ! [X: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F @ X @ Y4 ) )
=> ( ! [X: set_a,Y4: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X )
=> ( ( ord_less_eq_set_a @ Z4 @ X )
=> ( ord_less_eq_set_a @ ( F @ Y4 @ Z4 ) @ X ) ) )
=> ( ( sup_sup_set_a @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_926_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_927_sup_OorderI,axiom,
! [A2: filter_a,B2: filter_a] :
( ( A2
= ( sup_sup_filter_a @ A2 @ B2 ) )
=> ( ord_less_eq_filter_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_928_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_929_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_930_sup_OorderE,axiom,
! [B2: filter_a,A2: filter_a] :
( ( ord_less_eq_filter_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_filter_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_931_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_932_le__iff__sup,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_933_le__iff__sup,axiom,
( ord_less_eq_filter_a
= ( ^ [X3: filter_a,Y2: filter_a] :
( ( sup_sup_filter_a @ X3 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_934_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_935_sup__least,axiom,
! [Y3: set_set_a,X2: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ Z2 @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ Y3 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_936_sup__least,axiom,
! [Y3: filter_a,X2: filter_a,Z2: filter_a] :
( ( ord_less_eq_filter_a @ Y3 @ X2 )
=> ( ( ord_less_eq_filter_a @ Z2 @ X2 )
=> ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ Y3 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_937_sup__least,axiom,
! [Y3: set_a,X2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ Z2 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y3 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_938_sup__mono,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( sup_sup_set_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_939_sup__mono,axiom,
! [A2: filter_a,C: filter_a,B2: filter_a,D: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ C )
=> ( ( ord_less_eq_filter_a @ B2 @ D )
=> ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ A2 @ B2 ) @ ( sup_sup_filter_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_940_sup__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_941_sup_Omono,axiom,
! [C: set_set_a,A2: set_set_a,D: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ D @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ C @ D ) @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_942_sup_Omono,axiom,
! [C: filter_a,A2: filter_a,D: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ C @ A2 )
=> ( ( ord_less_eq_filter_a @ D @ B2 )
=> ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ C @ D ) @ ( sup_sup_filter_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_943_sup_Omono,axiom,
! [C: set_a,A2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_944_le__supI2,axiom,
! [X2: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_945_le__supI2,axiom,
! [X2: filter_a,B2: filter_a,A2: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ B2 )
=> ( ord_less_eq_filter_a @ X2 @ ( sup_sup_filter_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_946_le__supI2,axiom,
! [X2: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X2 @ B2 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_947_le__supI1,axiom,
! [X2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_948_le__supI1,axiom,
! [X2: filter_a,A2: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ X2 @ A2 )
=> ( ord_less_eq_filter_a @ X2 @ ( sup_sup_filter_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_949_le__supI1,axiom,
! [X2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_950_sup__ge2,axiom,
! [Y3: set_set_a,X2: set_set_a] : ( ord_le3724670747650509150_set_a @ Y3 @ ( sup_sup_set_set_a @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_951_sup__ge2,axiom,
! [Y3: filter_a,X2: filter_a] : ( ord_less_eq_filter_a @ Y3 @ ( sup_sup_filter_a @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_952_sup__ge2,axiom,
! [Y3: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_953_sup__ge1,axiom,
! [X2: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_954_sup__ge1,axiom,
! [X2: filter_a,Y3: filter_a] : ( ord_less_eq_filter_a @ X2 @ ( sup_sup_filter_a @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_955_sup__ge1,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_956_le__supI,axiom,
! [A2: set_set_a,X2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_957_le__supI,axiom,
! [A2: filter_a,X2: filter_a,B2: filter_a] :
( ( ord_less_eq_filter_a @ A2 @ X2 )
=> ( ( ord_less_eq_filter_a @ B2 @ X2 )
=> ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_958_le__supI,axiom,
! [A2: set_a,X2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X2 )
=> ( ( ord_less_eq_set_a @ B2 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_959_le__supE,axiom,
! [A2: set_set_a,B2: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ X2 )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_960_le__supE,axiom,
! [A2: filter_a,B2: filter_a,X2: filter_a] :
( ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_filter_a @ A2 @ X2 )
=> ~ ( ord_less_eq_filter_a @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_961_le__supE,axiom,
! [A2: set_a,B2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X2 )
=> ~ ( ord_less_eq_set_a @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_962_inf__sup__ord_I3_J,axiom,
! [X2: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_963_inf__sup__ord_I3_J,axiom,
! [X2: filter_a,Y3: filter_a] : ( ord_less_eq_filter_a @ X2 @ ( sup_sup_filter_a @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_964_inf__sup__ord_I3_J,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_965_inf__sup__ord_I4_J,axiom,
! [Y3: set_set_a,X2: set_set_a] : ( ord_le3724670747650509150_set_a @ Y3 @ ( sup_sup_set_set_a @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_966_inf__sup__ord_I4_J,axiom,
! [Y3: filter_a,X2: filter_a] : ( ord_less_eq_filter_a @ Y3 @ ( sup_sup_filter_a @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_967_inf__sup__ord_I4_J,axiom,
! [Y3: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_968_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_969_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_970_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_971_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_972_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( A4
= ( sup_sup_set_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_973_sup_Ostrict__order__iff,axiom,
( ord_less_set_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( A4
= ( sup_sup_set_set_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_974_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_975_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_976_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_977_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_978_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_979_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_980_less__supI2,axiom,
! [X2: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ X2 @ B2 )
=> ( ord_less_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_981_less__supI2,axiom,
! [X2: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ X2 @ B2 )
=> ( ord_less_set_set_a @ X2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_982_less__supI1,axiom,
! [X2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ X2 @ A2 )
=> ( ord_less_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_983_less__supI1,axiom,
! [X2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ X2 @ A2 )
=> ( ord_less_set_set_a @ X2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_984_sup__inf__distrib2,axiom,
! [Y3: set_a,Z2: set_a,X2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ Z2 ) @ X2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ X2 ) @ ( sup_sup_set_a @ Z2 @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_985_sup__inf__distrib2,axiom,
! [Y3: set_set_a,Z2: set_set_a,X2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y3 @ Z2 ) @ X2 )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y3 @ X2 ) @ ( sup_sup_set_set_a @ Z2 @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_986_sup__inf__distrib1,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_987_sup__inf__distrib1,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y3 ) @ ( sup_sup_set_set_a @ X2 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_988_inf__sup__distrib2,axiom,
! [Y3: set_a,Z2: set_a,X2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ Z2 ) @ X2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ X2 ) @ ( inf_inf_set_a @ Z2 @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_989_inf__sup__distrib2,axiom,
! [Y3: set_set_a,Z2: set_set_a,X2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y3 @ Z2 ) @ X2 )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y3 @ X2 ) @ ( inf_inf_set_set_a @ Z2 @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_990_inf__sup__distrib1,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_991_inf__sup__distrib1,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ ( inf_inf_set_set_a @ X2 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_992_distrib__imp2,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ! [X: set_a,Y4: set_a,Z4: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y4 @ Z4 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y4 ) @ ( sup_sup_set_a @ X @ Z4 ) ) )
=> ( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_993_distrib__imp2,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ! [X: set_set_a,Y4: set_set_a,Z4: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( inf_inf_set_set_a @ Y4 @ Z4 ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X @ Y4 ) @ ( sup_sup_set_set_a @ X @ Z4 ) ) )
=> ( ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ ( inf_inf_set_set_a @ X2 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_994_distrib__imp1,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ! [X: set_a,Y4: set_a,Z4: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y4 @ Z4 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y4 ) @ ( inf_inf_set_a @ X @ Z4 ) ) )
=> ( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_995_distrib__imp1,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ! [X: set_set_a,Y4: set_set_a,Z4: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( sup_sup_set_set_a @ Y4 @ Z4 ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X @ Y4 ) @ ( inf_inf_set_set_a @ X @ Z4 ) ) )
=> ( ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y3 ) @ ( sup_sup_set_set_a @ X2 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_996_distrib__sup__le,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y3 @ Z2 ) ) @ ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y3 ) @ ( sup_sup_set_set_a @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_997_distrib__sup__le,axiom,
! [X2: filter_a,Y3: filter_a,Z2: filter_a] : ( ord_less_eq_filter_a @ ( sup_sup_filter_a @ X2 @ ( inf_inf_filter_a @ Y3 @ Z2 ) ) @ ( inf_inf_filter_a @ ( sup_sup_filter_a @ X2 @ Y3 ) @ ( sup_sup_filter_a @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_998_distrib__sup__le,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_999_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: set_a,Z2: set_a,X2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ Z2 ) @ X2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ X2 ) @ ( sup_sup_set_a @ Z2 @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1000_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: set_set_a,Z2: set_set_a,X2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y3 @ Z2 ) @ X2 )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y3 @ X2 ) @ ( sup_sup_set_set_a @ Z2 @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1001_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: set_a,Z2: set_a,X2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ Z2 ) @ X2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ X2 ) @ ( inf_inf_set_a @ Z2 @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1002_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: set_set_a,Z2: set_set_a,X2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y3 @ Z2 ) @ X2 )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y3 @ X2 ) @ ( inf_inf_set_set_a @ Z2 @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1003_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1004_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y3 ) @ ( sup_sup_set_set_a @ X2 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1005_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1006_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y3 ) @ ( inf_inf_set_set_a @ X2 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1007_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_1008_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_1009_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_1010_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_1011_all__not__in__conv,axiom,
! [A: set_set_set_a] :
( ( ! [X3: set_set_a] :
~ ( member_set_set_a @ X3 @ A ) )
= ( A = bot_bo3380559777022489994_set_a ) ) ).
% all_not_in_conv
thf(fact_1012_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_1013_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_1014_empty__iff,axiom,
! [C: set_set_a] :
~ ( member_set_set_a @ C @ bot_bo3380559777022489994_set_a ) ).
% empty_iff
thf(fact_1015_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_1016_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_1017_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_1018_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_1019_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_1020_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_1021_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1022_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X2 )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1023_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1024_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1025_inf__bot__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_1026_inf__bot__right,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_1027_inf__bot__right,axiom,
! [X2: filter_a] :
( ( inf_inf_filter_a @ X2 @ bot_bot_filter_a )
= bot_bot_filter_a ) ).
% inf_bot_right
thf(fact_1028_inf__bot__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_1029_inf__bot__left,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X2 )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_1030_inf__bot__left,axiom,
! [X2: filter_a] :
( ( inf_inf_filter_a @ bot_bot_filter_a @ X2 )
= bot_bot_filter_a ) ).
% inf_bot_left
thf(fact_1031_open__empty,axiom,
topolo8477419352202985285open_a @ bot_bot_set_a ).
% open_empty
thf(fact_1032_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_1033_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_1034_sup__bot_Oright__neutral,axiom,
! [A2: filter_a] :
( ( sup_sup_filter_a @ A2 @ bot_bot_filter_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_1035_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_1036_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_1037_sup__bot_Oneutr__eq__iff,axiom,
! [A2: filter_a,B2: filter_a] :
( ( bot_bot_filter_a
= ( sup_sup_filter_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_filter_a )
& ( B2 = bot_bot_filter_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1038_closed__empty,axiom,
topolo784654279908865136osed_a @ bot_bot_set_a ).
% closed_empty
thf(fact_1039_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_1040_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_1041_atLeastatMost__empty_H,axiom,
! [A2: set_a,B2: set_a] :
( ~ ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( set_or6288561110385358355_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% atLeastatMost_empty'
thf(fact_1042_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_1043_atLeastLessThan__empty,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( set_or2348907005316661231_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_1044_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_1045_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_1046_greaterThanLessThan__empty,axiom,
! [L: set_a,K: set_a] :
( ( ord_less_eq_set_a @ L @ K )
=> ( ( set_or6017932776736107018_set_a @ K @ L )
= bot_bot_set_set_a ) ) ).
% greaterThanLessThan_empty
thf(fact_1047_greaterThanAtMost__empty,axiom,
! [L: set_a,K: set_a] :
( ( ord_less_eq_set_a @ L @ K )
=> ( ( set_or2503527069484367278_set_a @ K @ L )
= bot_bot_set_set_a ) ) ).
% greaterThanAtMost_empty
thf(fact_1048_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_1049_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_1050_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_1051_equals0I,axiom,
! [A: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_1052_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_1053_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_1054_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_1055_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_1056_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_1057_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1058_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_a @ X @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_1059_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_1060_separation__t2,axiom,
! [X2: a,Y3: a] :
( ( X2 != Y3 )
= ( ? [U: set_a,V: set_a] :
( ( topolo8477419352202985285open_a @ U )
& ( topolo8477419352202985285open_a @ V )
& ( member_a @ X2 @ U )
& ( member_a @ Y3 @ V )
& ( ( inf_inf_set_a @ U @ V )
= bot_bot_set_a ) ) ) ) ).
% separation_t2
thf(fact_1061_hausdorff,axiom,
! [X2: a,Y3: a] :
( ( X2 != Y3 )
=> ? [U2: set_a,V2: set_a] :
( ( topolo8477419352202985285open_a @ U2 )
& ( topolo8477419352202985285open_a @ V2 )
& ( member_a @ X2 @ U2 )
& ( member_a @ Y3 @ V2 )
& ( ( inf_inf_set_a @ U2 @ V2 )
= bot_bot_set_a ) ) ) ).
% hausdorff
thf(fact_1062_ivl__disj__int__one_I8_J,axiom,
! [L: a,U3: a] :
( ( inf_inf_set_a @ ( set_or5139330845457685135Than_a @ L @ U3 ) @ ( set_ord_atLeast_a @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(8)
thf(fact_1063_ivl__disj__int__one_I6_J,axiom,
! [L: a,U3: a] :
( ( inf_inf_set_a @ ( set_or5939364468397584554Than_a @ L @ U3 ) @ ( set_ord_atLeast_a @ U3 ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(6)
thf(fact_1064_subset__emptyI,axiom,
! [A: set_a] :
( ! [X: a] :
~ ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1065_ivl__disj__un__singleton_I4_J,axiom,
! [L: a,U3: a] :
( ( ord_less_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ U3 ) @ ( insert_a @ U3 @ bot_bot_set_a ) )
= ( set_or4472690218693186638Most_a @ L @ U3 ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_1066_ivl__disj__un__singleton_I3_J,axiom,
! [L: a,U3: a] :
( ( ord_less_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( insert_a @ L @ bot_bot_set_a ) @ ( set_or5939364468397584554Than_a @ L @ U3 ) )
= ( set_or5139330845457685135Than_a @ L @ U3 ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_1067_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_1068_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_1069_insert__subset,axiom,
! [X2: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( ( member_a @ X2 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_1070_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_closed__singleton,axiom,
! [A2: a] : ( topolo784654279908865136osed_a @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% closed_singleton
thf(fact_1082_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_1083_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_1084_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1085_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_1086_subset__insert,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_1087_insert__mono,axiom,
! [C2: set_a,D2: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_1088_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_1089_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_1090_closed__insert,axiom,
! [S: set_a,A2: a] :
( ( topolo784654279908865136osed_a @ S )
=> ( topolo784654279908865136osed_a @ ( insert_a @ A2 @ S ) ) ) ).
% closed_insert
thf(fact_1091_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_1092_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_1093_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_1094_Set_Oset__insert,axiom,
! [X2: a,A: set_a] :
( ( member_a @ X2 @ A )
=> ~ ! [B8: set_a] :
( ( A
= ( insert_a @ X2 @ B8 ) )
=> ( member_a @ X2 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_1095_insert__ident,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ~ ( member_a @ X2 @ B )
=> ( ( ( insert_a @ X2 @ A )
= ( insert_a @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1096_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1097_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_1098_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B8: set_a] :
( ( A
= ( insert_a @ A2 @ B8 ) )
& ~ ( member_a @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_1099_insert__subsetI,axiom,
! [X2: a,A: set_a,X5: set_a] :
( ( member_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ X5 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1100_subset__singletonD,axiom,
! [A: set_a,X2: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1101_subset__singleton__iff,axiom,
! [X5: set_a,A2: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1102_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_1103_connected__closed,axiom,
( topolo2370605967727889109cted_a
= ( ^ [S4: set_a] :
~ ? [A5: set_a,B5: set_a] :
( ( topolo784654279908865136osed_a @ A5 )
& ( topolo784654279908865136osed_a @ B5 )
& ( ord_less_eq_set_a @ S4 @ ( sup_sup_set_a @ A5 @ B5 ) )
& ( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ S4 )
= bot_bot_set_a )
& ( ( inf_inf_set_a @ A5 @ S4 )
!= bot_bot_set_a )
& ( ( inf_inf_set_a @ B5 @ S4 )
!= bot_bot_set_a ) ) ) ) ).
% connected_closed
thf(fact_1104_connectedD__interval,axiom,
! [U5: set_a,X2: a,Y3: a,Z2: a] :
( ( topolo2370605967727889109cted_a @ U5 )
=> ( ( member_a @ X2 @ U5 )
=> ( ( member_a @ Y3 @ U5 )
=> ( ( ord_less_eq_a @ X2 @ Z2 )
=> ( ( ord_less_eq_a @ Z2 @ Y3 )
=> ( member_a @ Z2 @ U5 ) ) ) ) ) ) ).
% connectedD_interval
thf(fact_1105_connected__contains__Icc,axiom,
! [A: set_a,A2: a,B2: a] :
( ( topolo2370605967727889109cted_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ A )
=> ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ A ) ) ) ) ).
% connected_contains_Icc
thf(fact_1106_connected__contains__Ioo,axiom,
! [A: set_a,A2: a,B2: a] :
( ( topolo2370605967727889109cted_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ A )
=> ( ord_less_eq_set_a @ ( set_or5939364468397584554Than_a @ A2 @ B2 ) @ A ) ) ) ) ).
% connected_contains_Ioo
thf(fact_1107_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X: a,Y4: a] :
( ( member_a @ X @ A )
=> ( ( member_a @ Y4 @ A )
=> ( X = Y4 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_1108_connectedD,axiom,
! [A: set_a,U5: set_a,V3: set_a] :
( ( topolo2370605967727889109cted_a @ A )
=> ( ( topolo8477419352202985285open_a @ U5 )
=> ( ( topolo8477419352202985285open_a @ V3 )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ U5 @ V3 ) @ A )
= bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ U5 @ V3 ) )
=> ( ( ( inf_inf_set_a @ U5 @ A )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ V3 @ A )
= bot_bot_set_a ) ) ) ) ) ) ) ).
% connectedD
thf(fact_1109_connectedI,axiom,
! [U5: set_a] :
( ! [A8: set_a] :
( ( topolo8477419352202985285open_a @ A8 )
=> ! [B8: set_a] :
( ( topolo8477419352202985285open_a @ B8 )
=> ( ( ( inf_inf_set_a @ A8 @ U5 )
!= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ B8 @ U5 )
!= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ A8 @ B8 ) @ U5 )
= bot_bot_set_a )
=> ~ ( ord_less_eq_set_a @ U5 @ ( sup_sup_set_a @ A8 @ B8 ) ) ) ) ) ) )
=> ( topolo2370605967727889109cted_a @ U5 ) ) ).
% connectedI
thf(fact_1110_connected__def,axiom,
( topolo2370605967727889109cted_a
= ( ^ [S2: set_a] :
~ ? [A5: set_a,B5: set_a] :
( ( topolo8477419352202985285open_a @ A5 )
& ( topolo8477419352202985285open_a @ B5 )
& ( ord_less_eq_set_a @ S2 @ ( sup_sup_set_a @ A5 @ B5 ) )
& ( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ S2 )
= bot_bot_set_a )
& ( ( inf_inf_set_a @ A5 @ S2 )
!= bot_bot_set_a )
& ( ( inf_inf_set_a @ B5 @ S2 )
!= bot_bot_set_a ) ) ) ) ).
% connected_def
thf(fact_1111_connected__closedD,axiom,
! [S3: set_a,A: set_a,B: set_a] :
( ( topolo2370605967727889109cted_a @ S3 )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ S3 )
= bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ S3 @ ( sup_sup_set_a @ A @ B ) )
=> ( ( topolo784654279908865136osed_a @ A )
=> ( ( topolo784654279908865136osed_a @ B )
=> ( ( ( inf_inf_set_a @ A @ S3 )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ S3 )
= bot_bot_set_a ) ) ) ) ) ) ) ).
% connected_closedD
thf(fact_1112_connected__closed__set,axiom,
! [S: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo2370605967727889109cted_a @ S )
= ( ~ ? [A5: set_a,B5: set_a] :
( ( topolo784654279908865136osed_a @ A5 )
& ( topolo784654279908865136osed_a @ B5 )
& ( A5 != bot_bot_set_a )
& ( B5 != bot_bot_set_a )
& ( ( sup_sup_set_a @ A5 @ B5 )
= S )
& ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a ) ) ) ) ) ).
% connected_closed_set
thf(fact_1113_connected__as__closed__union,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( topolo2370605967727889109cted_a @ C2 )
=> ( ( C2
= ( sup_sup_set_a @ A @ B ) )
=> ( ( topolo784654279908865136osed_a @ A )
=> ( ( topolo784654279908865136osed_a @ B )
=> ( ( A != bot_bot_set_a )
=> ( ( B != bot_bot_set_a )
=> ( ( inf_inf_set_a @ A @ B )
!= bot_bot_set_a ) ) ) ) ) ) ) ).
% connected_as_closed_union
thf(fact_1114_connected__diff__open__from__closed,axiom,
! [S3: set_a,T4: set_a,U3: set_a] :
( ( ord_less_eq_set_a @ S3 @ T4 )
=> ( ( ord_less_eq_set_a @ T4 @ U3 )
=> ( ( topolo8477419352202985285open_a @ S3 )
=> ( ( topolo784654279908865136osed_a @ T4 )
=> ( ( topolo2370605967727889109cted_a @ U3 )
=> ( ( topolo2370605967727889109cted_a @ ( minus_minus_set_a @ T4 @ S3 ) )
=> ( topolo2370605967727889109cted_a @ ( minus_minus_set_a @ U3 @ S3 ) ) ) ) ) ) ) ) ).
% connected_diff_open_from_closed
thf(fact_1115_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_1116_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_1117_insert__Diff1,axiom,
! [X2: a,B: set_a,A: set_a] :
( ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1118_Diff__insert0,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1119_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_1120_closed__Diff,axiom,
! [S: set_a,T5: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo8477419352202985285open_a @ T5 )
=> ( topolo784654279908865136osed_a @ ( minus_minus_set_a @ S @ T5 ) ) ) ) ).
% closed_Diff
thf(fact_1121_open__Diff,axiom,
! [S: set_a,T5: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( topolo784654279908865136osed_a @ T5 )
=> ( topolo8477419352202985285open_a @ ( minus_minus_set_a @ S @ T5 ) ) ) ) ).
% open_Diff
thf(fact_1122_double__diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1123_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1124_Diff__mono,axiom,
! [A: set_a,C2: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1125_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_1126_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_1127_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_1128_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1129_insert__Diff__if,axiom,
! [X2: a,B: set_a,A: set_a] :
( ( ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( insert_a @ X2 @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1130_diff__shunt__var,axiom,
! [X2: set_a,Y3: set_a] :
( ( ( minus_minus_set_a @ X2 @ Y3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1131_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X2: a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1132_Diff__insert__absorb,axiom,
! [X2: a,A: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1133_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_1134_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_1135_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_1136_subset__insert__iff,axiom,
! [A: set_a,X2: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( ( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1137_Diff__single__insert,axiom,
! [A: set_a,X2: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1138_psubset__insert__iff,axiom,
! [A: set_a,X2: a,B: set_a] :
( ( ord_less_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( ( ( member_a @ X2 @ B )
=> ( ord_less_set_a @ A @ B ) )
& ( ~ ( member_a @ X2 @ B )
=> ( ( ( member_a @ X2 @ A )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1139_open__delete,axiom,
! [S3: set_a,X2: a] :
( ( topolo8477419352202985285open_a @ S3 )
=> ( topolo8477419352202985285open_a @ ( minus_minus_set_a @ S3 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% open_delete
thf(fact_1140_at__within__nhd,axiom,
! [X2: a,S: set_a,T5: set_a,U5: set_a] :
( ( member_a @ X2 @ S )
=> ( ( topolo8477419352202985285open_a @ S )
=> ( ( ( minus_minus_set_a @ ( inf_inf_set_a @ T5 @ S ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ U5 @ S ) @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
=> ( ( topolo1902352237885396414thin_a @ X2 @ T5 )
= ( topolo1902352237885396414thin_a @ X2 @ U5 ) ) ) ) ) ).
% at_within_nhd
thf(fact_1141_ivl__disj__un__one_I6_J,axiom,
! [L: a,U3: a] :
( ( ord_less_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ U3 ) @ ( set_ord_atLeast_a @ U3 ) )
= ( set_or8632414552788122084Than_a @ L ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_1142_member__remove,axiom,
! [X2: a,Y3: a,A: set_a] :
( ( member_a @ X2 @ ( remove_a @ Y3 @ A ) )
= ( ( member_a @ X2 @ A )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_1143_greaterThan__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_or8632414552788122084Than_a @ K ) )
= ( ord_less_a @ K @ I2 ) ) ).
% greaterThan_iff
thf(fact_1144_greaterThan__subset__iff,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ X2 ) @ ( set_or8632414552788122084Than_a @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% greaterThan_subset_iff
thf(fact_1145_open__greaterThan,axiom,
! [A2: a] : ( topolo8477419352202985285open_a @ ( set_or8632414552788122084Than_a @ A2 ) ) ).
% open_greaterThan
thf(fact_1146_at__le,axiom,
! [S3: set_a,T4: set_a,X2: a] :
( ( ord_less_eq_set_a @ S3 @ T4 )
=> ( ord_less_eq_filter_a @ ( topolo1902352237885396414thin_a @ X2 @ S3 ) @ ( topolo1902352237885396414thin_a @ X2 @ T4 ) ) ) ).
% at_le
thf(fact_1147_at__within__Icc__at__right,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( topolo1902352237885396414thin_a @ A2 @ ( set_or672772299803893939Most_a @ A2 @ B2 ) )
= ( topolo1902352237885396414thin_a @ A2 @ ( set_or8632414552788122084Than_a @ A2 ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_1148_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_1149_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_1150_ivl__disj__un__one_I7_J,axiom,
! [L: a,U3: a] :
( ( ord_less_eq_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ U3 ) @ ( set_or8632414552788122084Than_a @ U3 ) )
= ( set_ord_atLeast_a @ L ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_1151_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_1152_at__eq__bot__iff,axiom,
! [A2: a] :
( ( ( topolo1902352237885396414thin_a @ A2 @ top_top_set_a )
= bot_bot_filter_a )
= ( topolo8477419352202985285open_a @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% at_eq_bot_iff
thf(fact_1153_eventually__at__rightI,axiom,
! [A2: a,B2: a,P: a > $o] :
( ! [X: a] :
( ( member_a @ X @ ( set_or5939364468397584554Than_a @ A2 @ B2 ) )
=> ( P @ X ) )
=> ( ( ord_less_a @ A2 @ B2 )
=> ( eventually_a @ P @ ( topolo1902352237885396414thin_a @ A2 @ ( set_or8632414552788122084Than_a @ A2 ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_1154_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_1155_open__UNIV,axiom,
topolo8477419352202985285open_a @ top_top_set_a ).
% open_UNIV
thf(fact_1156_lessThan__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_lessThan_a @ K ) )
= ( ord_less_a @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_1157_closed__UNIV,axiom,
topolo784654279908865136osed_a @ top_top_set_a ).
% closed_UNIV
thf(fact_1158_lessThan__subset__iff,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_lessThan_a @ X2 ) @ ( set_ord_lessThan_a @ Y3 ) )
= ( ord_less_eq_a @ X2 @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_1159_eventually__at__left,axiom,
! [Y3: a,X2: a,P: a > $o] :
( ( ord_less_a @ Y3 @ X2 )
=> ( ( eventually_a @ P @ ( topolo1902352237885396414thin_a @ X2 @ ( set_ord_lessThan_a @ X2 ) ) )
= ( ? [B4: a] :
( ( ord_less_a @ B4 @ X2 )
& ! [Y2: a] :
( ( ord_less_a @ B4 @ Y2 )
=> ( ( ord_less_a @ Y2 @ X2 )
=> ( P @ Y2 ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_1160_eventually__at__leftI,axiom,
! [A2: a,B2: a,P: a > $o] :
( ! [X: a] :
( ( member_a @ X @ ( set_or5939364468397584554Than_a @ A2 @ B2 ) )
=> ( P @ X ) )
=> ( ( ord_less_a @ A2 @ B2 )
=> ( eventually_a @ P @ ( topolo1902352237885396414thin_a @ B2 @ ( set_ord_lessThan_a @ B2 ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_1161_UNIV__witness,axiom,
? [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_witness
thf(fact_1162_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X: a] : ( member_a @ X @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_1163_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_1164_open__lessThan,axiom,
! [A2: a] : ( topolo8477419352202985285open_a @ ( set_ord_lessThan_a @ A2 ) ) ).
% open_lessThan
thf(fact_1165_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_1166_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_1167_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_1168_eventually__at__topological,axiom,
! [P: a > $o,A2: a,S3: set_a] :
( ( eventually_a @ P @ ( topolo1902352237885396414thin_a @ A2 @ S3 ) )
= ( ? [S2: set_a] :
( ( topolo8477419352202985285open_a @ S2 )
& ( member_a @ A2 @ S2 )
& ! [X3: a] :
( ( member_a @ X3 @ S2 )
=> ( ( X3 != A2 )
=> ( ( member_a @ X3 @ S3 )
=> ( P @ X3 ) ) ) ) ) ) ) ).
% eventually_at_topological
thf(fact_1169_lessThan__strict__subset__iff,axiom,
! [M3: a,N: a] :
( ( ord_less_set_a @ ( set_ord_lessThan_a @ M3 ) @ ( set_ord_lessThan_a @ N ) )
= ( ord_less_a @ M3 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_1170_at__within__open,axiom,
! [A2: a,S: set_a] :
( ( member_a @ A2 @ S )
=> ( ( topolo8477419352202985285open_a @ S )
=> ( ( topolo1902352237885396414thin_a @ A2 @ S )
= ( topolo1902352237885396414thin_a @ A2 @ top_top_set_a ) ) ) ) ).
% at_within_open
thf(fact_1171_at__within__Icc__at__left,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( topolo1902352237885396414thin_a @ B2 @ ( set_or672772299803893939Most_a @ A2 @ B2 ) )
= ( topolo1902352237885396414thin_a @ B2 @ ( set_ord_lessThan_a @ B2 ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_1172_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_1173_eventually__at__right,axiom,
! [X2: a,Y3: a,P: a > $o] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ( eventually_a @ P @ ( topolo1902352237885396414thin_a @ X2 @ ( set_or8632414552788122084Than_a @ X2 ) ) )
= ( ? [B4: a] :
( ( ord_less_a @ X2 @ B4 )
& ! [Y2: a] :
( ( ord_less_a @ X2 @ Y2 )
=> ( ( ord_less_a @ Y2 @ B4 )
=> ( P @ Y2 ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_1174_atLeastLessThan__def,axiom,
( set_or5139330845457685135Than_a
= ( ^ [L3: a,U4: a] : ( inf_inf_set_a @ ( set_ord_atLeast_a @ L3 ) @ ( set_ord_lessThan_a @ U4 ) ) ) ) ).
% atLeastLessThan_def
thf(fact_1175_at__within__Icc__at,axiom,
! [A2: a,X2: a,B2: a] :
( ( ord_less_a @ A2 @ X2 )
=> ( ( ord_less_a @ X2 @ B2 )
=> ( ( topolo1902352237885396414thin_a @ X2 @ ( set_or672772299803893939Most_a @ A2 @ B2 ) )
= ( topolo1902352237885396414thin_a @ X2 @ top_top_set_a ) ) ) ) ).
% at_within_Icc_at
thf(fact_1176_at__within__open__subset,axiom,
! [A2: a,S: set_a,T5: set_a] :
( ( member_a @ A2 @ S )
=> ( ( topolo8477419352202985285open_a @ S )
=> ( ( ord_less_eq_set_a @ S @ T5 )
=> ( ( topolo1902352237885396414thin_a @ A2 @ T5 )
= ( topolo1902352237885396414thin_a @ A2 @ top_top_set_a ) ) ) ) ) ).
% at_within_open_subset
thf(fact_1177_Iio__Int__singleton,axiom,
! [X2: a,K: a] :
( ( ( ord_less_a @ X2 @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= ( insert_a @ X2 @ bot_bot_set_a ) ) )
& ( ~ ( ord_less_a @ X2 @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= bot_bot_set_a ) ) ) ).
% Iio_Int_singleton
thf(fact_1178_less__separate,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ? [A3: a,B3: a] :
( ( member_a @ X2 @ ( set_ord_lessThan_a @ A3 ) )
& ( member_a @ Y3 @ ( set_or8632414552788122084Than_a @ B3 ) )
& ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ A3 ) @ ( set_or8632414552788122084Than_a @ B3 ) )
= bot_bot_set_a ) ) ) ).
% less_separate
thf(fact_1179_ivl__disj__un__one_I1_J,axiom,
! [L: a,U3: a] :
( ( ord_less_a @ L @ U3 )
=> ( ( sup_sup_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or5939364468397584554Than_a @ L @ U3 ) )
= ( set_ord_lessThan_a @ U3 ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_1180_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2496817875450240012_set_a @ inf_inf_set_a @ top_top_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1181_ComplI,axiom,
! [C: a,A: set_a] :
( ~ ( member_a @ C @ A )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A ) ) ) ).
% ComplI
thf(fact_1182_Compl__iff,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
= ( ~ ( member_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1183_compl__le__compl__iff,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X2 ) @ ( uminus_uminus_set_a @ Y3 ) )
= ( ord_less_eq_set_a @ Y3 @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_1184_Compl__anti__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B ) @ ( uminus_uminus_set_a @ A ) ) ) ).
% Compl_anti_mono
thf(fact_1185_Compl__subset__Compl__iff,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) )
= ( ord_less_eq_set_a @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_1186_closed__Compl,axiom,
! [S: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( topolo784654279908865136osed_a @ ( uminus_uminus_set_a @ S ) ) ) ).
% closed_Compl
thf(fact_1187_open__Compl,axiom,
! [S: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( topolo8477419352202985285open_a @ ( uminus_uminus_set_a @ S ) ) ) ).
% open_Compl
thf(fact_1188_Compl__lessThan,axiom,
! [K: a] :
( ( uminus_uminus_set_a @ ( set_ord_lessThan_a @ K ) )
= ( set_ord_atLeast_a @ K ) ) ).
% Compl_lessThan
thf(fact_1189_Compl__atLeast,axiom,
! [K: a] :
( ( uminus_uminus_set_a @ ( set_ord_atLeast_a @ K ) )
= ( set_ord_lessThan_a @ K ) ) ).
% Compl_atLeast
thf(fact_1190_subset__Compl__singleton,axiom,
! [A: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1191_compl__le__swap2,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y3 ) @ X2 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X2 ) @ Y3 ) ) ).
% compl_le_swap2
thf(fact_1192_compl__le__swap1,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ ( uminus_uminus_set_a @ X2 ) )
=> ( ord_less_eq_set_a @ X2 @ ( uminus_uminus_set_a @ Y3 ) ) ) ).
% compl_le_swap1
thf(fact_1193_compl__mono,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y3 ) @ ( uminus_uminus_set_a @ X2 ) ) ) ).
% compl_mono
thf(fact_1194_ComplD,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
=> ~ ( member_a @ C @ A ) ) ).
% ComplD
thf(fact_1195_closed__open,axiom,
( topolo784654279908865136osed_a
= ( ^ [S2: set_a] : ( topolo8477419352202985285open_a @ ( uminus_uminus_set_a @ S2 ) ) ) ) ).
% closed_open
thf(fact_1196_open__closed,axiom,
( topolo8477419352202985285open_a
= ( ^ [S2: set_a] : ( topolo784654279908865136osed_a @ ( uminus_uminus_set_a @ S2 ) ) ) ) ).
% open_closed
thf(fact_1197_subset__Compl__self__eq,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= ( A = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_1198_inf__shunt,axiom,
! [X2: set_a,Y3: set_a] :
( ( ( inf_inf_set_a @ X2 @ Y3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X2 @ ( uminus_uminus_set_a @ Y3 ) ) ) ).
% inf_shunt
thf(fact_1199_sup__shunt,axiom,
! [X2: set_a,Y3: set_a] :
( ( ( sup_sup_set_a @ X2 @ Y3 )
= top_top_set_a )
= ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X2 ) @ Y3 ) ) ).
% sup_shunt
thf(fact_1200_sup__neg__inf,axiom,
! [P3: set_a,Q3: set_a,R: set_a] :
( ( ord_less_eq_set_a @ P3 @ ( sup_sup_set_a @ Q3 @ R ) )
= ( ord_less_eq_set_a @ ( inf_inf_set_a @ P3 @ ( uminus_uminus_set_a @ Q3 ) ) @ R ) ) ).
% sup_neg_inf
thf(fact_1201_shunt2,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ ( uminus_uminus_set_a @ Y3 ) ) @ Z2 )
= ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) ) ) ).
% shunt2
thf(fact_1202_shunt1,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ Z2 )
= ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ Y3 ) @ Z2 ) ) ) ).
% shunt1
thf(fact_1203_disjoint__eq__subset__Compl,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1204_top_Oordering__top__axioms,axiom,
ordering_top_set_a @ ord_less_eq_set_a @ ord_less_set_a @ top_top_set_a ).
% top.ordering_top_axioms
thf(fact_1205_joinable__connected__component__eq,axiom,
! [T5: set_a,S: set_a,X2: a,Y3: a] :
( ( topolo2370605967727889109cted_a @ T5 )
=> ( ( ord_less_eq_set_a @ T5 @ S )
=> ( ( ( inf_inf_set_a @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ X2 ) ) @ T5 )
!= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ Y3 ) ) @ T5 )
!= bot_bot_set_a )
=> ( ( collect_a @ ( connec6912146525016367596nent_a @ S @ X2 ) )
= ( collect_a @ ( connec6912146525016367596nent_a @ S @ Y3 ) ) ) ) ) ) ) ).
% joinable_connected_component_eq
thf(fact_1206_connected__componentI,axiom,
! [T5: set_a,S: set_a,X2: a,Y3: a] :
( ( topolo2370605967727889109cted_a @ T5 )
=> ( ( ord_less_eq_set_a @ T5 @ S )
=> ( ( member_a @ X2 @ T5 )
=> ( ( member_a @ Y3 @ T5 )
=> ( connec6912146525016367596nent_a @ S @ X2 @ Y3 ) ) ) ) ) ).
% connected_componentI
thf(fact_1207_connected__component__mono,axiom,
! [S: set_a,T5: set_a,X2: a] :
( ( ord_less_eq_set_a @ S @ T5 )
=> ( ord_less_eq_set_a @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ X2 ) ) @ ( collect_a @ ( connec6912146525016367596nent_a @ T5 @ X2 ) ) ) ) ).
% connected_component_mono
thf(fact_1208_connected__component__subset,axiom,
! [S: set_a,X2: a] : ( ord_less_eq_set_a @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ X2 ) ) @ S ) ).
% connected_component_subset
thf(fact_1209_connected__component__of__subset,axiom,
! [S: set_a,X2: a,Y3: a,T5: set_a] :
( ( connec6912146525016367596nent_a @ S @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ S @ T5 )
=> ( connec6912146525016367596nent_a @ T5 @ X2 @ Y3 ) ) ) ).
% connected_component_of_subset
thf(fact_1210_connected__component__intermediate__subset,axiom,
! [U5: set_a,A2: a,T5: set_a] :
( ( ord_less_eq_set_a @ ( collect_a @ ( connec6912146525016367596nent_a @ U5 @ A2 ) ) @ T5 )
=> ( ( ord_less_eq_set_a @ T5 @ U5 )
=> ( ( collect_a @ ( connec6912146525016367596nent_a @ T5 @ A2 ) )
= ( collect_a @ ( connec6912146525016367596nent_a @ U5 @ A2 ) ) ) ) ) ).
% connected_component_intermediate_subset
thf(fact_1211_closed__connected__component,axiom,
! [S: set_a,X2: a] :
( ( topolo784654279908865136osed_a @ S )
=> ( topolo784654279908865136osed_a @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ X2 ) ) ) ) ).
% closed_connected_component
thf(fact_1212_connected__component__overlap,axiom,
! [S: set_a,A2: a,B2: a] :
( ( ( inf_inf_set_a @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ A2 ) ) @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ B2 ) ) )
!= bot_bot_set_a )
= ( ( member_a @ A2 @ S )
& ( member_a @ B2 @ S )
& ( ( collect_a @ ( connec6912146525016367596nent_a @ S @ A2 ) )
= ( collect_a @ ( connec6912146525016367596nent_a @ S @ B2 ) ) ) ) ) ).
% connected_component_overlap
thf(fact_1213_connected__component__disjoint,axiom,
! [S: set_a,A2: a,B2: a] :
( ( ( inf_inf_set_a @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ A2 ) ) @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ B2 ) ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ B2 ) ) ) ) ) ).
% connected_component_disjoint
thf(fact_1214_connected__component__nonoverlap,axiom,
! [S: set_a,A2: a,B2: a] :
( ( ( inf_inf_set_a @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ A2 ) ) @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ B2 ) ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ S )
| ~ ( member_a @ B2 @ S )
| ( ( collect_a @ ( connec6912146525016367596nent_a @ S @ A2 ) )
!= ( collect_a @ ( connec6912146525016367596nent_a @ S @ B2 ) ) ) ) ) ).
% connected_component_nonoverlap
thf(fact_1215_connected__component__maximal,axiom,
! [X2: a,T5: set_a,S: set_a] :
( ( member_a @ X2 @ T5 )
=> ( ( topolo2370605967727889109cted_a @ T5 )
=> ( ( ord_less_eq_set_a @ T5 @ S )
=> ( ord_less_eq_set_a @ T5 @ ( collect_a @ ( connec6912146525016367596nent_a @ S @ X2 ) ) ) ) ) ) ).
% connected_component_maximal
thf(fact_1216_connected__component__unique,axiom,
! [X2: a,C: set_a,S: set_a] :
( ( member_a @ X2 @ C )
=> ( ( ord_less_eq_set_a @ C @ S )
=> ( ( topolo2370605967727889109cted_a @ C )
=> ( ! [C5: set_a] :
( ( member_a @ X2 @ C5 )
=> ( ( ord_less_eq_set_a @ C5 @ S )
=> ( ( topolo2370605967727889109cted_a @ C5 )
=> ( ord_less_eq_set_a @ C5 @ C ) ) ) )
=> ( ( collect_a @ ( connec6912146525016367596nent_a @ S @ X2 ) )
= C ) ) ) ) ) ).
% connected_component_unique
thf(fact_1217_connected__component__def,axiom,
( connec6912146525016367596nent_a
= ( ^ [S2: set_a,X3: a,Y2: a] :
? [T2: set_a] :
( ( topolo2370605967727889109cted_a @ T2 )
& ( ord_less_eq_set_a @ T2 @ S2 )
& ( member_a @ X3 @ T2 )
& ( member_a @ Y2 @ T2 ) ) ) ) ).
% connected_component_def
thf(fact_1218_closed__Inter,axiom,
! [K3: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ K3 )
=> ( topolo784654279908865136osed_a @ X ) )
=> ( topolo784654279908865136osed_a @ ( comple6135023378680113637_set_a @ K3 ) ) ) ).
% closed_Inter
thf(fact_1219_cInf__atLeastAtMost,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ Y3 @ X2 ) )
= Y3 ) ) ).
% cInf_atLeastAtMost
thf(fact_1220_Inf__atLeastAtMost,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ X2 @ Y3 ) )
= X2 ) ) ).
% Inf_atLeastAtMost
thf(fact_1221_cInf__eq__non__empty,axiom,
! [X5: set_set_a,A2: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ X5 )
=> ( ord_less_eq_set_a @ A2 @ X ) )
=> ( ! [Y4: set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ X5 )
=> ( ord_less_eq_set_a @ Y4 @ X4 ) )
=> ( ord_less_eq_set_a @ Y4 @ A2 ) )
=> ( ( comple6135023378680113637_set_a @ X5 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1222_cInf__greatest,axiom,
! [X5: set_set_a,Z2: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ X5 )
=> ( ord_less_eq_set_a @ Z2 @ X ) )
=> ( ord_less_eq_set_a @ Z2 @ ( comple6135023378680113637_set_a @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_1223_cInf__eq__minimum,axiom,
! [Z2: set_a,X5: set_set_a] :
( ( member_set_a @ Z2 @ X5 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ X5 )
=> ( ord_less_eq_set_a @ Z2 @ X ) )
=> ( ( comple6135023378680113637_set_a @ X5 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_1224_Inter__lower,axiom,
! [B: set_a,A: set_set_a] :
( ( member_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ B ) ) ).
% Inter_lower
thf(fact_1225_Inter__greatest,axiom,
! [A: set_set_a,C2: set_a] :
( ! [X6: set_a] :
( ( member_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ C2 @ X6 ) )
=> ( ord_less_eq_set_a @ C2 @ ( comple6135023378680113637_set_a @ A ) ) ) ).
% Inter_greatest
thf(fact_1226_Inter__anti__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inter_anti_mono
thf(fact_1227_Inter__subset,axiom,
! [A: set_set_a,B: set_a] :
( ! [X6: set_a] :
( ( member_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ X6 @ B ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_1228_Inf__eqI,axiom,
! [A: set_set_a,X2: set_a] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A )
=> ( ord_less_eq_set_a @ X2 @ I3 ) )
=> ( ! [Y4: set_a] :
( ! [I4: set_a] :
( ( member_set_a @ I4 @ A )
=> ( ord_less_eq_set_a @ Y4 @ I4 ) )
=> ( ord_less_eq_set_a @ Y4 @ X2 ) )
=> ( ( comple6135023378680113637_set_a @ A )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_1229_Inf__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ! [B3: set_a] :
( ( member_set_a @ B3 @ B )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ X4 @ B3 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_mono
thf(fact_1230_Inf__lower,axiom,
! [X2: set_a,A: set_set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ X2 ) ) ).
% Inf_lower
thf(fact_1231_Inf__lower2,axiom,
! [U3: set_a,A: set_set_a,V4: set_a] :
( ( member_set_a @ U3 @ A )
=> ( ( ord_less_eq_set_a @ U3 @ V4 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ V4 ) ) ) ).
% Inf_lower2
thf(fact_1232_le__Inf__iff,axiom,
! [B2: set_a,A: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( comple6135023378680113637_set_a @ A ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_1233_Inf__greatest,axiom,
! [A: set_set_a,Z2: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ Z2 @ X ) )
=> ( ord_less_eq_set_a @ Z2 @ ( comple6135023378680113637_set_a @ A ) ) ) ).
% Inf_greatest
thf(fact_1234_Inter__Un__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B ) ) ) ).
% Inter_Un_subset
thf(fact_1235_Inf__superset__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_superset_mono
thf(fact_1236_Inf__less__eq,axiom,
! [A: set_set_a,U3: set_a] :
( ! [V5: set_a] :
( ( member_set_a @ V5 @ A )
=> ( ord_less_eq_set_a @ V5 @ U3 ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ U3 ) ) ) ).
% Inf_less_eq
thf(fact_1237_less__eq__Inf__inter,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B ) ) ) ).
% less_eq_Inf_inter
thf(fact_1238_chains__extend,axiom,
! [C: set_set_a,S: set_set_a,Z2: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ Z2 @ S )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ C )
=> ( ord_less_eq_set_a @ X @ Z2 ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z2 @ bot_bot_set_set_a ) @ C ) @ ( chains_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1239_chainsD,axiom,
! [C: set_set_a,S: set_set_a,X2: set_a,Y3: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ X2 @ C )
=> ( ( member_set_a @ Y3 @ C )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
| ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ) ).
% chainsD
thf(fact_1240_Zorn__Lemma2,axiom,
! [A: set_set_a] :
( ! [X: set_set_a] :
( ( member_set_set_a @ X @ ( chains_a @ A ) )
=> ? [Xa: set_a] :
( ( member_set_a @ Xa @ A )
& ! [Xb: set_a] :
( ( member_set_a @ Xb @ X )
=> ( ord_less_eq_set_a @ Xb @ Xa ) ) ) )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1241_less__eq__cInf__inter,axiom,
! [A: set_set_a,B: set_set_a] :
( ( condit8937546108433946286_set_a @ A )
=> ( ( condit8937546108433946286_set_a @ B )
=> ( ( ( inf_inf_set_set_a @ A @ B )
!= bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1242_bdd__belowI,axiom,
! [A: set_a,M3: a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_a @ M3 @ X ) )
=> ( condit5901475214736682318elow_a @ A ) ) ).
% bdd_belowI
thf(fact_1243_bdd__belowI,axiom,
! [A: set_set_a,M3: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ M3 @ X ) )
=> ( condit8937546108433946286_set_a @ A ) ) ).
% bdd_belowI
thf(fact_1244_bdd__below_OI,axiom,
! [A: set_a,M: a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_a @ M @ X ) )
=> ( condit5901475214736682318elow_a @ A ) ) ).
% bdd_below.I
thf(fact_1245_bdd__below_OI,axiom,
! [A: set_set_a,M: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ M @ X ) )
=> ( condit8937546108433946286_set_a @ A ) ) ).
% bdd_below.I
thf(fact_1246_bdd__below__Ici,axiom,
! [A2: a] : ( condit5901475214736682318elow_a @ ( set_ord_atLeast_a @ A2 ) ) ).
% bdd_below_Ici
thf(fact_1247_mono__atLeast,axiom,
! [B: a] : ( extend2808419353335425523_set_a @ ( set_ord_atLeast_a @ B ) ) ).
% mono_atLeast
thf(fact_1248_bdd__below__mono,axiom,
! [B: set_a,A: set_a] :
( ( condit5901475214736682318elow_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( condit5901475214736682318elow_a @ A ) ) ) ).
% bdd_below_mono
thf(fact_1249_mono__set,axiom,
( extend2808419353335425523_set_a
= ( ^ [S2: set_a] :
! [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( member_a @ X3 @ S2 )
=> ( member_a @ Y2 @ S2 ) ) ) ) ) ).
% mono_set
thf(fact_1250_mono__set,axiom,
( extend347329919781519187_set_a
= ( ^ [S2: set_set_a] :
! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( member_set_a @ X3 @ S2 )
=> ( member_set_a @ Y2 @ S2 ) ) ) ) ) ).
% mono_set
thf(fact_1251_bdd__below_OE,axiom,
! [A: set_a] :
( ( condit5901475214736682318elow_a @ A )
=> ~ ! [M4: a] :
~ ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ord_less_eq_a @ M4 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1252_bdd__below_OE,axiom,
! [A: set_set_a] :
( ( condit8937546108433946286_set_a @ A )
=> ~ ! [M4: set_a] :
~ ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ M4 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1253_bdd__below_Ounfold,axiom,
( condit8937546108433946286_set_a
= ( ^ [A5: set_set_a] :
? [M2: set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( ord_less_eq_set_a @ M2 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1254_cInf__lower,axiom,
! [X2: set_a,X5: set_set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ( condit8937546108433946286_set_a @ X5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X5 ) @ X2 ) ) ) ).
% cInf_lower
thf(fact_1255_cInf__lower2,axiom,
! [X2: set_a,X5: set_set_a,Y3: set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( condit8937546108433946286_set_a @ X5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X5 ) @ Y3 ) ) ) ) ).
% cInf_lower2
thf(fact_1256_le__cInf__iff,axiom,
! [S: set_set_a,A2: set_a] :
( ( S != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ S )
=> ( ( ord_less_eq_set_a @ A2 @ ( comple6135023378680113637_set_a @ S ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ S )
=> ( ord_less_eq_set_a @ A2 @ X3 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1257_cInf__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( B != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ A )
=> ( ! [B3: set_a] :
( ( member_set_a @ B3 @ B )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ X4 @ B3 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ) ) ).
% cInf_mono
thf(fact_1258_cInf__superset__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ B ) @ ( comple6135023378680113637_set_a @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1259_image__eqI,axiom,
! [B2: a,F: a > a,X2: a,A: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1260_bdd__belowI2,axiom,
! [A: set_a,M3: set_a,F: a > set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ M3 @ ( F @ X ) ) )
=> ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1261_bdd__below_OI2,axiom,
! [A: set_a,M: set_a,F: a > set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ M @ ( F @ X ) ) )
=> ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1262_cINF__lower,axiom,
! [F: a > set_a,A: set_a,X2: a] :
( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) )
=> ( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_1263_cINF__lower2,axiom,
! [F: a > set_a,A: set_a,X2: a,U3: set_a] :
( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) )
=> ( ( member_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ ( F @ X2 ) @ U3 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) ) @ U3 ) ) ) ) ).
% cINF_lower2
thf(fact_1264_INF__eq__iff,axiom,
! [I5: set_a,F: a > set_a,C: set_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_less_eq_set_a @ ( F @ I3 ) @ C ) )
=> ( ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ I5 ) )
= C )
= ( ! [X3: a] :
( ( member_a @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1265_INF__eq,axiom,
! [A: set_a,B: set_a,G: a > set_a,F: a > set_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) )
= ( comple6135023378680113637_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1266_cINF__greatest,axiom,
! [A: set_a,M3: set_a,F: a > set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ M3 @ ( F @ X ) ) )
=> ( ord_less_eq_set_a @ M3 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1267_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1268_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A )
=> ( B
!= ( image_a_a @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_1269_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1270_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_1271_in__image__insert__iff,axiom,
! [B: set_set_a,X2: a,A: set_a] :
( ! [C6: set_a] :
( ( member_set_a @ C6 @ B )
=> ~ ( member_a @ X2 @ C6 ) )
=> ( ( member_set_a @ A @ ( image_set_a_set_a @ ( insert_a @ X2 ) @ B ) )
= ( ( member_a @ X2 @ A )
& ( member_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1272_imageI,axiom,
! [X2: a,A: set_a,F: a > a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_1273_rev__image__eqI,axiom,
! [X2: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
% Helper facts (3)
thf(help_If_3_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X2: set_a,Y3: set_a] :
( ( if_set_a @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X2: set_a,Y3: set_a] :
( ( if_set_a @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [T: set_a] :
( ( topolo8477419352202985285open_a @ T )
& ( member_a @ x @ T )
& ( ord_less_eq_set_a @ T @ i ) ) ).
%------------------------------------------------------------------------------