TPTP Problem File: SLH0189^1.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 : LP_Duality/0000_Minimum_Maximum/prob_00053_001552__28699158_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1415 ( 610 unt; 137 typ; 0 def)
% Number of atoms : 3861 (1213 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10463 ( 303 ~; 66 |; 223 &;8327 @)
% ( 0 <=>;1544 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 1289 (1289 >; 0 *; 0 +; 0 <<)
% Number of symbols : 134 ( 131 usr; 15 con; 0-4 aty)
% Number of variables : 3569 ( 185 ^;3315 !; 69 ?;3569 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:52:34.410
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
set_o_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 (131)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_Itf__a_M_Eo_J,type,
complete_Inf_Inf_a_o: set_a_o > a > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple9105089376463352645_set_a: set_set_set_a > set_set_a ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__a_J,type,
comple6135023378680113637_set_a: set_set_a > set_a ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001tf__a,type,
condit5901475214736682318elow_a: set_a > $o ).
thf(sy_c_Fun_Omonotone__on_001_062_I_Eo_Mtf__a_J_001_062_I_Eo_Mtf__a_J,type,
monotone_on_o_a_o_a: set_o_a > ( ( $o > a ) > ( $o > a ) > $o ) > ( ( $o > a ) > ( $o > a ) > $o ) > ( ( $o > a ) > $o > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001_062_I_Eo_Mtf__a_J_001t__Set__Oset_Itf__a_J,type,
monoto888385181794500650_set_a: set_o_a > ( ( $o > a ) > ( $o > a ) > $o ) > ( set_a > set_a > $o ) > ( ( $o > a ) > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001_062_I_Eo_Mtf__a_J_001tf__a,type,
monotone_on_o_a_a: set_o_a > ( ( $o > a ) > ( $o > a ) > $o ) > ( a > a > $o ) > ( ( $o > a ) > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001_062_I_Eo_Mtf__a_J,type,
monoto9194799565806279554_a_o_a: set_set_a > ( set_a > set_a > $o ) > ( ( $o > a ) > ( $o > a ) > $o ) > ( set_a > $o > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
monoto7172710143293369831_set_a: set_set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > ( set_a > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001tf__a,type,
monotone_on_set_a_a: set_set_a > ( set_a > set_a > $o ) > ( a > a > $o ) > ( set_a > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001_062_I_Eo_Mtf__a_J,type,
monotone_on_a_o_a: set_a > ( a > a > $o ) > ( ( $o > a ) > ( $o > a ) > $o ) > ( a > $o > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001t__Set__Oset_Itf__a_J,type,
monotone_on_a_set_a: set_a > ( a > a > $o ) > ( set_a > set_a > $o ) > ( a > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001tf__a,type,
monotone_on_a_a: set_a > ( a > a > $o ) > ( a > a > $o ) > ( a > a ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__a_M_Eo_J,type,
uminus_uminus_a_o: ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
uminus6103902357914783669_set_a: set_set_a > set_set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_Itf__a_J,type,
comple3341859861669737308_set_a: ( set_a > set_a ) > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
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_Minimum__Maximum_OMaximum_001tf__a,type,
minimum_Maximum_a: set_a > a ).
thf(sy_c_Minimum__Maximum_OMinimum_001tf__a,type,
minimum_Minimum_a: set_a > a ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001_062_I_Eo_Mtf__a_J,type,
minimu315547183909508560um_o_a: set_o_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001t__Set__Oset_Itf__a_J,type,
minimu8775777210878807577_set_a: set_set_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001tf__a,type,
minimu6197867597544231097imum_a: set_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001_062_I_Eo_Mtf__a_J,type,
minimu4657282916794952894um_o_a: set_o_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001t__Set__Oset_Itf__a_J,type,
minimu6896447672505010603_set_a: set_set_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001tf__a,type,
minimu7473987258551571531imum_a: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
bot_bot_set_o_a: set_o_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
bot_bot_set_a_o: set_a_o ).
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_OLeast_001_062_I_Eo_Mtf__a_J,type,
ord_Least_o_a: ( ( $o > a ) > $o ) > $o > a ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_Itf__a_J,type,
ord_Least_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Oord__class_OLeast_001tf__a,type,
ord_Least_a: ( a > $o ) > a ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_Mtf__a_J,type,
ord_less_o_a: ( $o > a ) > ( $o > a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
ord_less_set_o_a: set_o_a > set_o_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mtf__a_J_J,type,
ord_less_eq_o_o_a: ( $o > $o > a ) > ( $o > $o > a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mtf__a_J,type,
ord_less_eq_o_a: ( $o > a ) > ( $o > a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
ord_less_eq_set_o_a: set_o_a > set_o_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_001_062_I_Eo_Mtf__a_J,type,
order_Greatest_o_a: ( ( $o > a ) > $o ) > $o > 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_001_062_Itf__a_M_Eo_J,type,
ordering_top_a_o: ( ( a > $o ) > ( a > $o ) > $o ) > ( ( a > $o ) > ( a > $o ) > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
orderi5875812994216768367_set_a: ( set_set_a > set_set_a > $o ) > ( set_set_a > set_set_a > $o ) > set_set_a > $o ).
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_Opartial__preordering_001_062_I_Eo_Mtf__a_J,type,
partia5423788306336055317ng_o_a: ( ( $o > a ) > ( $o > a ) > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Set__Oset_Itf__a_J,type,
partia6602192050731689876_set_a: ( set_a > set_a > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001tf__a,type,
partia125584492769400372ring_a: ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
top_top_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
top_top_set_o_a: set_o_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
top_top_set_a_o2: set_a_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
top_to4027821306633060462_set_a: set_set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OBall_001t__Set__Oset_Itf__a_J,type,
ball_set_a: set_set_a > ( set_a > $o ) > $o ).
thf(sy_c_Set_OBall_001tf__a,type,
ball_a: set_a > ( a > $o ) > $o ).
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_001_062_I_Eo_Mtf__a_J_001_062_I_Eo_Mtf__a_J,type,
image_o_a_o_a: ( ( $o > a ) > $o > a ) > set_o_a > set_o_a ).
thf(sy_c_Set_Oimage_001_062_I_Eo_Mtf__a_J_001t__Set__Oset_Itf__a_J,type,
image_o_a_set_a: ( ( $o > a ) > set_a ) > set_o_a > set_set_a ).
thf(sy_c_Set_Oimage_001_062_I_Eo_Mtf__a_J_001tf__a,type,
image_o_a_a: ( ( $o > a ) > a ) > set_o_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_062_I_Eo_Mtf__a_J,type,
image_set_a_o_a: ( set_a > $o > a ) > set_set_a > set_o_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_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_I_Eo_Mtf__a_J,type,
image_a_o_a: ( a > $o > a ) > set_a > set_o_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__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_Itf__a_J,type,
is_singleton_set_a: set_set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Opairwise_001t__Set__Oset_Itf__a_J,type,
pairwise_set_a: ( set_a > set_a > $o ) > set_set_a > $o ).
thf(sy_c_Set_Opairwise_001tf__a,type,
pairwise_a: ( a > a > $o ) > set_a > $o ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__a_J,type,
the_elem_set_a: set_set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_062_I_Eo_Mtf__a_J,type,
set_or8441445163928040022st_o_a: ( $o > a ) > ( $o > a ) > set_o_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_062_Itf__a_M_Eo_J,type,
set_or497184483940929162st_a_o: ( a > $o ) > ( a > $o ) > set_a_o ).
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_001_062_I_Eo_Mtf__a_J,type,
set_or4510008498168808314an_o_a: ( $o > a ) > ( $o > a ) > set_o_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_Itf__a_J,type,
set_or2348907005316661231_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001tf__a,type,
set_or5139330845457685135Than_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001_062_I_Eo_Mtf__a_J,type,
set_ord_atLeast_o_a: ( $o > a ) > set_o_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_Itf__a_J,type,
set_or8362275514725411625_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001tf__a,type,
set_ord_atLeast_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_I_Eo_Mtf__a_J,type,
set_ord_atMost_o_a: ( $o > a ) > set_o_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_Itf__a_M_Eo_J,type,
set_ord_atMost_a_o: ( a > $o ) > set_a_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or4016371710855203973_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001tf__a,type,
set_ord_atMost_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001_062_I_Eo_Mtf__a_J,type,
set_or1794473594002085499st_o_a: ( $o > a ) > ( $o > a ) > set_o_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_Itf__a_J,type,
set_or2503527069484367278_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001tf__a,type,
set_or4472690218693186638Most_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001_062_I_Eo_Mtf__a_J,type,
set_or8643659207013295391an_o_a: ( $o > a ) > ( $o > a ) > set_o_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_Itf__a_J,type,
set_or6017932776736107018_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001tf__a,type,
set_or5939364468397584554Than_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_Itf__a_J,type,
set_or460448635090783044_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001tf__a,type,
set_or8632414552788122084Than_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_Itf__a_M_Eo_J,type,
set_ord_lessThan_a_o: ( a > $o ) > set_a_o ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or5369375139905502561_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_Itf__a_J,type,
set_or5421148953861284865_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001tf__a,type,
set_ord_lessThan_a: a > set_a ).
thf(sy_c_member_001_062_I_Eo_Mtf__a_J,type,
member_o_a: ( $o > a ) > set_o_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_Eo_J,type,
member_a_o: ( a > $o ) > set_a_o > $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_S,type,
s: set_a ).
thf(sy_v_m____,type,
m: a ).
% Relevant facts (1277)
thf(fact_0_assms,axiom,
minimu7473987258551571531imum_a @ s ).
% assms
thf(fact_1__C_K_C_I1_J,axiom,
member_a @ m @ s ).
% "*"(1)
thf(fact_2__C_K_C_I2_J,axiom,
! [Y: a] :
( ( member_a @ Y @ s )
=> ( ord_less_eq_a @ m @ Y ) ) ).
% "*"(2)
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_A_092_060lbrakk_062m_A_092_060in_062_AS_059_A_092_060And_062y_O_Ay_A_092_060in_062_AS_A_092_060Longrightarrow_062_Am_A_092_060le_062_Ay_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [M: a] :
( ( member_a @ M @ s )
=> ~ ! [Y2: a] :
( ( member_a @ Y2 @ s )
=> ( ord_less_eq_a @ M @ Y2 ) ) ) ).
% \<open>\<And>thesis. (\<And>m. \<lbrakk>m \<in> S; \<And>y. y \<in> S \<Longrightarrow> m \<le> y\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_eqMinimumI,axiom,
! [X: a,S: set_a] :
( ( member_a @ X @ S )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ S )
=> ( ord_less_eq_a @ X @ Y3 ) )
=> ( ( minimum_Minimum_a @ S )
= X ) ) ) ).
% eqMinimumI
thf(fact_5__092_060open_062_092_060exists_062x_O_Ax_A_092_060in_062_AS_A_092_060and_062_ABall_AS_A_I_I_092_060le_062_J_Ax_J_092_060close_062,axiom,
? [X2: a] :
( ( member_a @ X2 @ s )
& ! [Xa: a] :
( ( member_a @ Xa @ s )
=> ( ord_less_eq_a @ X2 @ Xa ) ) ) ).
% \<open>\<exists>x. x \<in> S \<and> Ball S ((\<le>) x)\<close>
thf(fact_6_has__Minimum__def,axiom,
( minimu6896447672505010603_set_a
= ( ^ [S2: set_set_a] :
? [X3: set_a] :
( ( member_set_a @ X3 @ S2 )
& ! [Y4: set_a] :
( ( member_set_a @ Y4 @ S2 )
=> ( ord_less_eq_set_a @ X3 @ Y4 ) ) ) ) ) ).
% has_Minimum_def
thf(fact_7_has__Minimum__def,axiom,
( minimu4657282916794952894um_o_a
= ( ^ [S2: set_o_a] :
? [X3: $o > a] :
( ( member_o_a @ X3 @ S2 )
& ! [Y4: $o > a] :
( ( member_o_a @ Y4 @ S2 )
=> ( ord_less_eq_o_a @ X3 @ Y4 ) ) ) ) ) ).
% has_Minimum_def
thf(fact_8_has__Minimum__def,axiom,
( minimu7473987258551571531imum_a
= ( ^ [S2: set_a] :
? [X3: a] :
( ( member_a @ X3 @ S2 )
& ! [Y4: a] :
( ( member_a @ Y4 @ S2 )
=> ( ord_less_eq_a @ X3 @ Y4 ) ) ) ) ) ).
% has_Minimum_def
thf(fact_9_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_10_order__refl,axiom,
! [X: $o > a] : ( ord_less_eq_o_a @ X @ X ) ).
% order_refl
thf(fact_11_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_12_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_13_dual__order_Orefl,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% dual_order.refl
thf(fact_14_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_15_has__Maximum__def,axiom,
( minimu8775777210878807577_set_a
= ( ^ [S2: set_set_a] :
? [X3: set_a] :
( ( member_set_a @ X3 @ S2 )
& ! [Y4: set_a] :
( ( member_set_a @ Y4 @ S2 )
=> ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_16_has__Maximum__def,axiom,
( minimu315547183909508560um_o_a
= ( ^ [S2: set_o_a] :
? [X3: $o > a] :
( ( member_o_a @ X3 @ S2 )
& ! [Y4: $o > a] :
( ( member_o_a @ Y4 @ S2 )
=> ( ord_less_eq_o_a @ Y4 @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_17_has__Maximum__def,axiom,
( minimu6197867597544231097imum_a
= ( ^ [S2: set_a] :
? [X3: a] :
( ( member_a @ X3 @ S2 )
& ! [Y4: a] :
( ( member_a @ Y4 @ S2 )
=> ( ord_less_eq_a @ Y4 @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_18_Ball__def,axiom,
( ball_set_a
= ( ^ [A2: set_set_a,P: set_a > $o] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( P @ X3 ) ) ) ) ).
% Ball_def
thf(fact_19_Ball__def,axiom,
( ball_a
= ( ^ [A2: set_a,P: a > $o] :
! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( P @ X3 ) ) ) ) ).
% Ball_def
thf(fact_20_ball__reg,axiom,
! [R: set_set_a,P2: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ R )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ R )
=> ( P2 @ X2 ) )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_21_ball__reg,axiom,
! [R: set_a,P2: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ R )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ R )
=> ( P2 @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_22_Ball__Collect,axiom,
( ball_a
= ( ^ [A2: set_a,P: a > $o] : ( ord_less_eq_set_a @ A2 @ ( collect_a @ P ) ) ) ) ).
% Ball_Collect
thf(fact_23_eqMaximumI,axiom,
! [X: a,S: set_a] :
( ( member_a @ X @ S )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ S )
=> ( ord_less_eq_a @ Y3 @ X ) )
=> ( ( minimum_Maximum_a @ S )
= X ) ) ) ).
% eqMaximumI
thf(fact_24_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_25_verit__comp__simplify1_I2_J,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_26_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_27_nle__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_eq_a @ A @ B ) )
= ( ( ord_less_eq_a @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_28_le__cases3,axiom,
! [X: a,Y: a,Z: a] :
( ( ( ord_less_eq_a @ X @ Y )
=> ~ ( ord_less_eq_a @ Y @ Z ) )
=> ( ( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_eq_a @ X @ Z ) )
=> ( ( ( ord_less_eq_a @ X @ Z )
=> ~ ( ord_less_eq_a @ Z @ Y ) )
=> ( ( ( ord_less_eq_a @ Z @ Y )
=> ~ ( ord_less_eq_a @ Y @ X ) )
=> ( ( ( ord_less_eq_a @ Y @ Z )
=> ~ ( ord_less_eq_a @ Z @ X ) )
=> ~ ( ( ord_less_eq_a @ Z @ X )
=> ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_29_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_30_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: $o > a,Z2: $o > a] : ( Y5 = Z2 ) )
= ( ^ [X3: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X3 @ Y4 )
& ( ord_less_eq_o_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_31_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 ) )
= ( ^ [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
& ( ord_less_eq_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_32_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_33_ord__eq__le__trans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( A = B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_34_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_35_subsetI,axiom,
! [A3: set_set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( member_set_a @ X2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A3 @ B2 ) ) ).
% subsetI
thf(fact_36_subsetI,axiom,
! [A3: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A3 @ B2 ) ) ).
% subsetI
thf(fact_37_subset__antisym,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_38_in__mono,axiom,
! [A3: set_set_a,B2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
=> ( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_39_in__mono,axiom,
! [A3: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_40_subsetD,axiom,
! [A3: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
=> ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_41_subsetD,axiom,
! [A3: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_42_equalityE,axiom,
! [A3: set_a,B2: set_a] :
( ( A3 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_43_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A2: set_set_a,B3: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_44_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_45_equalityD1,axiom,
! [A3: set_a,B2: set_a] :
( ( A3 = B2 )
=> ( ord_less_eq_set_a @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_46_equalityD2,axiom,
! [A3: set_a,B2: set_a] :
( ( A3 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_47_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A2: set_set_a,B3: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A2 )
=> ( member_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_48_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A2 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_49_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_50_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_51_subset__trans,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_52_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_53_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_54_has__MaximumD_I1_J,axiom,
! [S: set_a] :
( ( minimu6197867597544231097imum_a @ S )
=> ( member_a @ ( minimum_Maximum_a @ S ) @ S ) ) ).
% has_MaximumD(1)
thf(fact_55_has__MaximumD_I2_J,axiom,
! [S: set_a,X: a] :
( ( minimu6197867597544231097imum_a @ S )
=> ( ( member_a @ X @ S )
=> ( ord_less_eq_a @ X @ ( minimum_Maximum_a @ S ) ) ) ) ).
% has_MaximumD(2)
thf(fact_56_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_57_order__antisym__conv,axiom,
! [Y: $o > a,X: $o > a] :
( ( ord_less_eq_o_a @ Y @ X )
=> ( ( ord_less_eq_o_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_58_order__antisym__conv,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_59_linorder__le__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_60_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_61_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_62_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_63_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_64_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > $o > a,C: $o > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_65_ord__le__eq__subst,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_66_ord__le__eq__subst,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > set_a,C: set_a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_67_ord__le__eq__subst,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_68_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_69_ord__eq__le__subst,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_70_ord__eq__le__subst,axiom,
! [A: $o > a,F: a > $o > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_71_ord__eq__le__subst,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_72_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_73_ord__eq__le__subst,axiom,
! [A: $o > a,F: set_a > $o > a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_74_ord__eq__le__subst,axiom,
! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_75_ord__eq__le__subst,axiom,
! [A: set_a,F: ( $o > a ) > set_a,B: $o > a,C: $o > a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_76_ord__eq__le__subst,axiom,
! [A: $o > a,F: ( $o > a ) > $o > a,B: $o > a,C: $o > a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_77_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_78_linorder__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_linear
thf(fact_79_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_80_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_81_order__eq__refl,axiom,
! [X: $o > a,Y: $o > a] :
( ( X = Y )
=> ( ord_less_eq_o_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_82_order__eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_83_order__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_84_order__subst2,axiom,
! [A: a,B: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_85_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_86_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_87_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > $o > a,C: $o > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_88_order__subst2,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_89_order__subst2,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > set_a,C: set_a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_90_order__subst2,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_91_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_92_order__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_93_order__subst1,axiom,
! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_94_order__subst1,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_95_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_96_order__subst1,axiom,
! [A: set_a,F: ( $o > a ) > set_a,B: $o > a,C: $o > a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_97_order__subst1,axiom,
! [A: $o > a,F: a > $o > a,B: a,C: a] :
( ( ord_less_eq_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_98_order__subst1,axiom,
! [A: $o > a,F: set_a > $o > a,B: set_a,C: set_a] :
( ( ord_less_eq_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_99_order__subst1,axiom,
! [A: $o > a,F: ( $o > a ) > $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_100_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_101_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [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_102_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: $o > a,Z2: $o > a] : ( Y5 = Z2 ) )
= ( ^ [A4: $o > a,B4: $o > a] :
( ( ord_less_eq_o_a @ A4 @ B4 )
& ( ord_less_eq_o_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_103_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ( ord_less_eq_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_104_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_105_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_106_Collect__mem__eq,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_107_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_108_Collect__cong,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_109_le__fun__def,axiom,
( ord_less_eq_o_a
= ( ^ [F2: $o > a,G: $o > a] :
! [X3: $o] : ( ord_less_eq_a @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_110_le__funI,axiom,
! [F: $o > a,G2: $o > a] :
( ! [X2: $o] : ( ord_less_eq_a @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq_o_a @ F @ G2 ) ) ).
% le_funI
thf(fact_111_le__funE,axiom,
! [F: $o > a,G2: $o > a,X: $o] :
( ( ord_less_eq_o_a @ F @ G2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_112_le__funD,axiom,
! [F: $o > a,G2: $o > a,X: $o] :
( ( ord_less_eq_o_a @ F @ G2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funD
thf(fact_113_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_114_antisym,axiom,
! [A: $o > a,B: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_115_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_116_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_117_dual__order_Otrans,axiom,
! [B: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ C @ B )
=> ( ord_less_eq_o_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_118_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_119_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_120_dual__order_Oantisym,axiom,
! [B: $o > a,A: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_121_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_122_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [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_123_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: $o > a,Z2: $o > a] : ( Y5 = Z2 ) )
= ( ^ [A4: $o > a,B4: $o > a] :
( ( ord_less_eq_o_a @ B4 @ A4 )
& ( ord_less_eq_o_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_124_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ( ord_less_eq_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_125_linorder__wlog,axiom,
! [P2: a > a > $o,A: a,B: a] :
( ! [A5: a,B5: a] :
( ( ord_less_eq_a @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: a,B5: a] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_126_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_127_order__trans,axiom,
! [X: $o > a,Y: $o > a,Z: $o > a] :
( ( ord_less_eq_o_a @ X @ Y )
=> ( ( ord_less_eq_o_a @ Y @ Z )
=> ( ord_less_eq_o_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_128_order__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z )
=> ( ord_less_eq_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_129_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_130_order_Otrans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% order.trans
thf(fact_131_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_132_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_133_order__antisym,axiom,
! [X: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X @ Y )
=> ( ( ord_less_eq_o_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_134_order__antisym,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_135_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_136_ord__le__eq__trans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_137_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_138_Greatest__equality,axiom,
! [P2: set_a > $o,X: set_a] :
( ( P2 @ X )
=> ( ! [Y3: set_a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X ) )
=> ( ( order_Greatest_set_a @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_139_Greatest__equality,axiom,
! [P2: ( $o > a ) > $o,X: $o > a] :
( ( P2 @ X )
=> ( ! [Y3: $o > a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_o_a @ Y3 @ X ) )
=> ( ( order_Greatest_o_a @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_140_Greatest__equality,axiom,
! [P2: a > $o,X: a] :
( ( P2 @ X )
=> ( ! [Y3: a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X ) )
=> ( ( order_Greatest_a @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_141_GreatestI2__order,axiom,
! [P2: set_a > $o,X: set_a,Q: set_a > $o] :
( ( P2 @ X )
=> ( ! [Y3: set_a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X ) )
=> ( ! [X2: set_a] :
( ( P2 @ X2 )
=> ( ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_142_GreatestI2__order,axiom,
! [P2: ( $o > a ) > $o,X: $o > a,Q: ( $o > a ) > $o] :
( ( P2 @ X )
=> ( ! [Y3: $o > a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_o_a @ Y3 @ X ) )
=> ( ! [X2: $o > a] :
( ( P2 @ X2 )
=> ( ! [Y2: $o > a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o_a @ Y2 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_o_a @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_143_GreatestI2__order,axiom,
! [P2: a > $o,X: a,Q: a > $o] :
( ( P2 @ X )
=> ( ! [Y3: a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X ) )
=> ( ! [X2: a] :
( ( P2 @ X2 )
=> ( ! [Y2: a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_a @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_144_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X5: $o > set_a,Y6: $o > set_a] :
( ( ord_less_eq_set_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_145_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_a
= ( ^ [X5: $o > $o > a,Y6: $o > $o > a] :
( ( ord_less_eq_o_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_146_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_a
= ( ^ [X5: $o > a,Y6: $o > a] :
( ( ord_less_eq_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_147_order_Opartial__preordering__axioms,axiom,
partia6602192050731689876_set_a @ ord_less_eq_set_a ).
% order.partial_preordering_axioms
thf(fact_148_order_Opartial__preordering__axioms,axiom,
partia5423788306336055317ng_o_a @ ord_less_eq_o_a ).
% order.partial_preordering_axioms
thf(fact_149_order_Opartial__preordering__axioms,axiom,
partia125584492769400372ring_a @ ord_less_eq_a ).
% order.partial_preordering_axioms
thf(fact_150_greaterThan__subset__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ X ) @ ( set_or8632414552788122084Than_a @ Y ) )
= ( ord_less_eq_a @ Y @ X ) ) ).
% greaterThan_subset_iff
thf(fact_151_lessThan__subset__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_set_a @ ( set_ord_lessThan_a @ X ) @ ( set_ord_lessThan_a @ Y ) )
= ( ord_less_eq_a @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_152_atLeast__subset__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or8362275514725411625_set_a @ X ) @ ( set_or8362275514725411625_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_153_atLeast__subset__iff,axiom,
! [X: $o > a,Y: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_ord_atLeast_o_a @ X ) @ ( set_ord_atLeast_o_a @ Y ) )
= ( ord_less_eq_o_a @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_154_atLeast__subset__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ X ) @ ( set_ord_atLeast_a @ Y ) )
= ( ord_less_eq_a @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_155_atMost__subset__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X ) @ ( set_ord_atMost_set_a @ Y ) )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_156_atMost__subset__iff,axiom,
! [X: $o > a,Y: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_ord_atMost_o_a @ X ) @ ( set_ord_atMost_o_a @ Y ) )
= ( ord_less_eq_o_a @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_157_atMost__subset__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ X ) @ ( set_ord_atMost_a @ Y ) )
= ( ord_less_eq_a @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_158_pairwise__mono,axiom,
! [P2: a > a > $o,A3: set_a,Q: a > a > $o,B2: set_a] :
( ( pairwise_a @ P2 @ A3 )
=> ( ! [X2: a,Y3: a] :
( ( P2 @ X2 @ Y3 )
=> ( Q @ X2 @ Y3 ) )
=> ( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( pairwise_a @ Q @ B2 ) ) ) ) ).
% pairwise_mono
thf(fact_159_pairwise__subset,axiom,
! [P2: a > a > $o,S: set_a,T2: set_a] :
( ( pairwise_a @ P2 @ S )
=> ( ( ord_less_eq_set_a @ T2 @ S )
=> ( pairwise_a @ P2 @ T2 ) ) ) ).
% pairwise_subset
thf(fact_160_atMost__eq__iff,axiom,
! [X: a,Y: a] :
( ( ( set_ord_atMost_a @ X )
= ( set_ord_atMost_a @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_161_lessThan__eq__iff,axiom,
! [X: a,Y: a] :
( ( ( set_ord_lessThan_a @ X )
= ( set_ord_lessThan_a @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_162_atLeast__eq__iff,axiom,
! [X: a,Y: a] :
( ( ( set_ord_atLeast_a @ X )
= ( set_ord_atLeast_a @ Y ) )
= ( X = Y ) ) ).
% atLeast_eq_iff
thf(fact_163_greaterThan__eq__iff,axiom,
! [X: a,Y: a] :
( ( ( set_or8632414552788122084Than_a @ X )
= ( set_or8632414552788122084Than_a @ Y ) )
= ( X = Y ) ) ).
% greaterThan_eq_iff
thf(fact_164_atMost__iff,axiom,
! [I: set_a,K: set_a] :
( ( member_set_a @ I @ ( set_ord_atMost_set_a @ K ) )
= ( ord_less_eq_set_a @ I @ K ) ) ).
% atMost_iff
thf(fact_165_atMost__iff,axiom,
! [I: $o > a,K: $o > a] :
( ( member_o_a @ I @ ( set_ord_atMost_o_a @ K ) )
= ( ord_less_eq_o_a @ I @ K ) ) ).
% atMost_iff
thf(fact_166_atMost__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_ord_atMost_a @ K ) )
= ( ord_less_eq_a @ I @ K ) ) ).
% atMost_iff
thf(fact_167_atLeast__iff,axiom,
! [I: set_a,K: set_a] :
( ( member_set_a @ I @ ( set_or8362275514725411625_set_a @ K ) )
= ( ord_less_eq_set_a @ K @ I ) ) ).
% atLeast_iff
thf(fact_168_atLeast__iff,axiom,
! [I: $o > a,K: $o > a] :
( ( member_o_a @ I @ ( set_ord_atLeast_o_a @ K ) )
= ( ord_less_eq_o_a @ K @ I ) ) ).
% atLeast_iff
thf(fact_169_atLeast__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_ord_atLeast_a @ K ) )
= ( ord_less_eq_a @ K @ I ) ) ).
% atLeast_iff
thf(fact_170_Ioi__le__Ico,axiom,
! [A: a] : ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ A ) @ ( set_ord_atLeast_a @ A ) ) ).
% Ioi_le_Ico
thf(fact_171_partial__preordering__def,axiom,
( partia125584492769400372ring_a
= ( ^ [Less_eq: a > a > $o] :
( ! [A4: a] : ( Less_eq @ A4 @ A4 )
& ! [A4: a,B4: a,C3: a] :
( ( Less_eq @ A4 @ B4 )
=> ( ( Less_eq @ B4 @ C3 )
=> ( Less_eq @ A4 @ C3 ) ) ) ) ) ) ).
% partial_preordering_def
thf(fact_172_pairwise__def,axiom,
( pairwise_a
= ( ^ [R2: a > a > $o,S2: set_a] :
! [X3: a] :
( ( member_a @ X3 @ S2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ S2 )
=> ( ( X3 != Y4 )
=> ( R2 @ X3 @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_173_pairwiseI,axiom,
! [S: set_set_a,R: set_a > set_a > $o] :
( ! [X2: set_a,Y3: set_a] :
( ( member_set_a @ X2 @ S )
=> ( ( member_set_a @ Y3 @ S )
=> ( ( X2 != Y3 )
=> ( R @ X2 @ Y3 ) ) ) )
=> ( pairwise_set_a @ R @ S ) ) ).
% pairwiseI
thf(fact_174_pairwiseI,axiom,
! [S: set_a,R: a > a > $o] :
( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ S )
=> ( ( member_a @ Y3 @ S )
=> ( ( X2 != Y3 )
=> ( R @ X2 @ Y3 ) ) ) )
=> ( pairwise_a @ R @ S ) ) ).
% pairwiseI
thf(fact_175_pairwiseD,axiom,
! [R: set_a > set_a > $o,S: set_set_a,X: set_a,Y: set_a] :
( ( pairwise_set_a @ R @ S )
=> ( ( member_set_a @ X @ S )
=> ( ( member_set_a @ Y @ S )
=> ( ( X != Y )
=> ( R @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_176_pairwiseD,axiom,
! [R: a > a > $o,S: set_a,X: a,Y: a] :
( ( pairwise_a @ R @ S )
=> ( ( member_a @ X @ S )
=> ( ( member_a @ Y @ S )
=> ( ( X != Y )
=> ( R @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_177_partial__preordering_Otrans,axiom,
! [Less_eq2: a > a > $o,A: a,B: a,C: a] :
( ( partia125584492769400372ring_a @ Less_eq2 )
=> ( ( Less_eq2 @ A @ B )
=> ( ( Less_eq2 @ B @ C )
=> ( Less_eq2 @ A @ C ) ) ) ) ).
% partial_preordering.trans
thf(fact_178_partial__preordering_Ointro,axiom,
! [Less_eq2: a > a > $o] :
( ! [A5: a] : ( Less_eq2 @ A5 @ A5 )
=> ( ! [A5: a,B5: a,C4: a] :
( ( Less_eq2 @ A5 @ B5 )
=> ( ( Less_eq2 @ B5 @ C4 )
=> ( Less_eq2 @ A5 @ C4 ) ) )
=> ( partia125584492769400372ring_a @ Less_eq2 ) ) ) ).
% partial_preordering.intro
thf(fact_179_partial__preordering_Orefl,axiom,
! [Less_eq2: a > a > $o,A: a] :
( ( partia125584492769400372ring_a @ Less_eq2 )
=> ( Less_eq2 @ A @ A ) ) ).
% partial_preordering.refl
thf(fact_180_Least1I,axiom,
! [P2: set_a > $o] :
( ? [X4: set_a] :
( ( P2 @ X4 )
& ! [Y3: set_a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_set_a @ X4 @ Y3 ) )
& ! [Y3: set_a] :
( ( ( P2 @ Y3 )
& ! [Ya: set_a] :
( ( P2 @ Ya )
=> ( ord_less_eq_set_a @ Y3 @ Ya ) ) )
=> ( Y3 = X4 ) ) )
=> ( P2 @ ( ord_Least_set_a @ P2 ) ) ) ).
% Least1I
thf(fact_181_Least1I,axiom,
! [P2: ( $o > a ) > $o] :
( ? [X4: $o > a] :
( ( P2 @ X4 )
& ! [Y3: $o > a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_o_a @ X4 @ Y3 ) )
& ! [Y3: $o > a] :
( ( ( P2 @ Y3 )
& ! [Ya: $o > a] :
( ( P2 @ Ya )
=> ( ord_less_eq_o_a @ Y3 @ Ya ) ) )
=> ( Y3 = X4 ) ) )
=> ( P2 @ ( ord_Least_o_a @ P2 ) ) ) ).
% Least1I
thf(fact_182_Least1I,axiom,
! [P2: a > $o] :
( ? [X4: a] :
( ( P2 @ X4 )
& ! [Y3: a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_a @ X4 @ Y3 ) )
& ! [Y3: a] :
( ( ( P2 @ Y3 )
& ! [Ya: a] :
( ( P2 @ Ya )
=> ( ord_less_eq_a @ Y3 @ Ya ) ) )
=> ( Y3 = X4 ) ) )
=> ( P2 @ ( ord_Least_a @ P2 ) ) ) ).
% Least1I
thf(fact_183_Least1__le,axiom,
! [P2: set_a > $o,Z: set_a] :
( ? [X4: set_a] :
( ( P2 @ X4 )
& ! [Y3: set_a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_set_a @ X4 @ Y3 ) )
& ! [Y3: set_a] :
( ( ( P2 @ Y3 )
& ! [Ya: set_a] :
( ( P2 @ Ya )
=> ( ord_less_eq_set_a @ Y3 @ Ya ) ) )
=> ( Y3 = X4 ) ) )
=> ( ( P2 @ Z )
=> ( ord_less_eq_set_a @ ( ord_Least_set_a @ P2 ) @ Z ) ) ) ).
% Least1_le
thf(fact_184_Least1__le,axiom,
! [P2: ( $o > a ) > $o,Z: $o > a] :
( ? [X4: $o > a] :
( ( P2 @ X4 )
& ! [Y3: $o > a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_o_a @ X4 @ Y3 ) )
& ! [Y3: $o > a] :
( ( ( P2 @ Y3 )
& ! [Ya: $o > a] :
( ( P2 @ Ya )
=> ( ord_less_eq_o_a @ Y3 @ Ya ) ) )
=> ( Y3 = X4 ) ) )
=> ( ( P2 @ Z )
=> ( ord_less_eq_o_a @ ( ord_Least_o_a @ P2 ) @ Z ) ) ) ).
% Least1_le
thf(fact_185_Least1__le,axiom,
! [P2: a > $o,Z: a] :
( ? [X4: a] :
( ( P2 @ X4 )
& ! [Y3: a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_a @ X4 @ Y3 ) )
& ! [Y3: a] :
( ( ( P2 @ Y3 )
& ! [Ya: a] :
( ( P2 @ Ya )
=> ( ord_less_eq_a @ Y3 @ Ya ) ) )
=> ( Y3 = X4 ) ) )
=> ( ( P2 @ Z )
=> ( ord_less_eq_a @ ( ord_Least_a @ P2 ) @ Z ) ) ) ).
% Least1_le
thf(fact_186_LeastI2__order,axiom,
! [P2: set_a > $o,X: set_a,Q: set_a > $o] :
( ( P2 @ X )
=> ( ! [Y3: set_a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_set_a @ X @ Y3 ) )
=> ( ! [X2: set_a] :
( ( P2 @ X2 )
=> ( ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_set_a @ P2 ) ) ) ) ) ).
% LeastI2_order
thf(fact_187_LeastI2__order,axiom,
! [P2: ( $o > a ) > $o,X: $o > a,Q: ( $o > a ) > $o] :
( ( P2 @ X )
=> ( ! [Y3: $o > a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_o_a @ X @ Y3 ) )
=> ( ! [X2: $o > a] :
( ( P2 @ X2 )
=> ( ! [Y2: $o > a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o_a @ X2 @ Y2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_o_a @ P2 ) ) ) ) ) ).
% LeastI2_order
thf(fact_188_LeastI2__order,axiom,
! [P2: a > $o,X: a,Q: a > $o] :
( ( P2 @ X )
=> ( ! [Y3: a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_a @ X @ Y3 ) )
=> ( ! [X2: a] :
( ( P2 @ X2 )
=> ( ! [Y2: a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_a @ P2 ) ) ) ) ) ).
% LeastI2_order
thf(fact_189_Least__equality,axiom,
! [P2: set_a > $o,X: set_a] :
( ( P2 @ X )
=> ( ! [Y3: set_a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_set_a @ X @ Y3 ) )
=> ( ( ord_Least_set_a @ P2 )
= X ) ) ) ).
% Least_equality
thf(fact_190_Least__equality,axiom,
! [P2: ( $o > a ) > $o,X: $o > a] :
( ( P2 @ X )
=> ( ! [Y3: $o > a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_o_a @ X @ Y3 ) )
=> ( ( ord_Least_o_a @ P2 )
= X ) ) ) ).
% Least_equality
thf(fact_191_Least__equality,axiom,
! [P2: a > $o,X: a] :
( ( P2 @ X )
=> ( ! [Y3: a] :
( ( P2 @ Y3 )
=> ( ord_less_eq_a @ X @ Y3 ) )
=> ( ( ord_Least_a @ P2 )
= X ) ) ) ).
% Least_equality
thf(fact_192_Ici__subset__Ioi__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ A ) @ ( set_or8632414552788122084Than_a @ B ) )
= ( ord_less_a @ B @ A ) ) ).
% Ici_subset_Ioi_iff
thf(fact_193_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_194_Icc__subset__Ici__iff,axiom,
! [L: $o > a,H: $o > a,L2: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_or8441445163928040022st_o_a @ L @ H ) @ ( set_ord_atLeast_o_a @ L2 ) )
= ( ~ ( ord_less_eq_o_a @ L @ H )
| ( ord_less_eq_o_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_195_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_196_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_197_Icc__subset__Iic__iff,axiom,
! [L: $o > a,H: $o > a,H2: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_or8441445163928040022st_o_a @ L @ H ) @ ( set_ord_atMost_o_a @ H2 ) )
= ( ~ ( ord_less_eq_o_a @ L @ H )
| ( ord_less_eq_o_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_198_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_199_Iic__subset__Iio__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ A ) @ ( set_ord_lessThan_a @ B ) )
= ( ord_less_a @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_200_Compl__atMost,axiom,
! [K: a] :
( ( uminus_uminus_set_a @ ( set_ord_atMost_a @ K ) )
= ( set_or8632414552788122084Than_a @ K ) ) ).
% Compl_atMost
thf(fact_201_Compl__greaterThan,axiom,
! [K: a] :
( ( uminus_uminus_set_a @ ( set_or8632414552788122084Than_a @ K ) )
= ( set_ord_atMost_a @ K ) ) ).
% Compl_greaterThan
thf(fact_202_Compl__atLeast,axiom,
! [K: a] :
( ( uminus_uminus_set_a @ ( set_ord_atLeast_a @ K ) )
= ( set_ord_lessThan_a @ K ) ) ).
% Compl_atLeast
thf(fact_203_Compl__lessThan,axiom,
! [K: a] :
( ( uminus_uminus_set_a @ ( set_ord_lessThan_a @ K ) )
= ( set_ord_atLeast_a @ K ) ) ).
% Compl_lessThan
thf(fact_204_ivl__subset,axiom,
! [I: a,J: a,M2: a,N: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ I @ J ) @ ( set_or5139330845457685135Than_a @ M2 @ N ) )
= ( ( ord_less_eq_a @ J @ I )
| ( ( ord_less_eq_a @ M2 @ I )
& ( ord_less_eq_a @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_205_top__apply,axiom,
( top_top_a_o
= ( ^ [X3: a] : top_top_o ) ) ).
% top_apply
thf(fact_206_atMost__UNIV__triv,axiom,
( ( set_ord_atMost_set_a @ top_top_set_a )
= top_top_set_set_a ) ).
% atMost_UNIV_triv
thf(fact_207_atMost__UNIV__triv,axiom,
( ( set_or4016371710855203973_set_a @ top_top_set_set_a )
= top_to4027821306633060462_set_a ) ).
% atMost_UNIV_triv
thf(fact_208_UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_209_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_210_psubsetI,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_set_a @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_211_ComplI,axiom,
! [C: set_a,A3: set_set_a] :
( ~ ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A3 ) ) ) ).
% ComplI
thf(fact_212_ComplI,axiom,
! [C: a,A3: set_a] :
( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% ComplI
thf(fact_213_Compl__iff,axiom,
! [C: set_a,A3: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A3 ) )
= ( ~ ( member_set_a @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_214_Compl__iff,axiom,
! [C: a,A3: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) )
= ( ~ ( member_a @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_215_Compl__eq__Compl__iff,axiom,
! [A3: set_a,B2: set_a] :
( ( ( uminus_uminus_set_a @ A3 )
= ( uminus_uminus_set_a @ B2 ) )
= ( A3 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_216_atLeastAtMost__iff,axiom,
! [I: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I @ ( set_or6288561110385358355_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I )
& ( ord_less_eq_set_a @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_217_atLeastAtMost__iff,axiom,
! [I: $o > a,L: $o > a,U: $o > a] :
( ( member_o_a @ I @ ( set_or8441445163928040022st_o_a @ L @ U ) )
= ( ( ord_less_eq_o_a @ L @ I )
& ( ord_less_eq_o_a @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_218_atLeastAtMost__iff,axiom,
! [I: a,L: a,U: a] :
( ( member_a @ I @ ( set_or672772299803893939Most_a @ L @ U ) )
= ( ( ord_less_eq_a @ L @ I )
& ( ord_less_eq_a @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_219_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_220_Icc__eq__Icc,axiom,
! [L: $o > a,H: $o > a,L2: $o > a,H2: $o > a] :
( ( ( set_or8441445163928040022st_o_a @ L @ H )
= ( set_or8441445163928040022st_o_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_o_a @ L @ H )
& ~ ( ord_less_eq_o_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_221_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_222_lessThan__iff,axiom,
! [I: set_a,K: set_a] :
( ( member_set_a @ I @ ( set_or5421148953861284865_set_a @ K ) )
= ( ord_less_set_a @ I @ K ) ) ).
% lessThan_iff
thf(fact_223_lessThan__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_ord_lessThan_a @ K ) )
= ( ord_less_a @ I @ K ) ) ).
% lessThan_iff
thf(fact_224_Compl__anti__mono,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B2 ) @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_225_Compl__subset__Compl__iff,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A3 ) @ ( uminus_uminus_set_a @ B2 ) )
= ( ord_less_eq_set_a @ B2 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_226_greaterThan__iff,axiom,
! [I: set_a,K: set_a] :
( ( member_set_a @ I @ ( set_or460448635090783044_set_a @ K ) )
= ( ord_less_set_a @ K @ I ) ) ).
% greaterThan_iff
thf(fact_227_greaterThan__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_or8632414552788122084Than_a @ K ) )
= ( ord_less_a @ K @ I ) ) ).
% greaterThan_iff
thf(fact_228_atLeastLessThan__iff,axiom,
! [I: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I @ ( set_or2348907005316661231_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I )
& ( ord_less_set_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_229_atLeastLessThan__iff,axiom,
! [I: $o > a,L: $o > a,U: $o > a] :
( ( member_o_a @ I @ ( set_or4510008498168808314an_o_a @ L @ U ) )
= ( ( ord_less_eq_o_a @ L @ I )
& ( ord_less_o_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_230_atLeastLessThan__iff,axiom,
! [I: a,L: a,U: a] :
( ( member_a @ I @ ( set_or5139330845457685135Than_a @ L @ U ) )
= ( ( ord_less_eq_a @ L @ I )
& ( ord_less_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_231_atLeastatMost__subset__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A @ B ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_232_atLeastatMost__subset__iff,axiom,
! [A: $o > a,B: $o > a,C: $o > a,D: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_or8441445163928040022st_o_a @ A @ B ) @ ( set_or8441445163928040022st_o_a @ C @ D ) )
= ( ~ ( ord_less_eq_o_a @ A @ B )
| ( ( ord_less_eq_o_a @ C @ A )
& ( ord_less_eq_o_a @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_233_atLeastatMost__subset__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A @ B ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_234_atLeastatMost__psubset__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_less_set_set_a @ ( set_or6288561110385358355_set_a @ A @ B ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_a @ A @ B )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B @ D )
& ( ( ord_less_set_a @ C @ A )
| ( ord_less_set_a @ B @ D ) ) ) )
& ( ord_less_eq_set_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_235_atLeastatMost__psubset__iff,axiom,
! [A: $o > a,B: $o > a,C: $o > a,D: $o > a] :
( ( ord_less_set_o_a @ ( set_or8441445163928040022st_o_a @ A @ B ) @ ( set_or8441445163928040022st_o_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_o_a @ A @ B )
| ( ( ord_less_eq_o_a @ C @ A )
& ( ord_less_eq_o_a @ B @ D )
& ( ( ord_less_o_a @ C @ A )
| ( ord_less_o_a @ B @ D ) ) ) )
& ( ord_less_eq_o_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_236_atLeastatMost__psubset__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_set_a @ ( set_or672772299803893939Most_a @ A @ B ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_a @ A @ B )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D )
& ( ( ord_less_a @ C @ A )
| ( ord_less_a @ B @ D ) ) ) )
& ( ord_less_eq_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_237_atLeastLessThan__inj_I2_J,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( set_or5139330845457685135Than_a @ A @ B )
= ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_238_atLeastLessThan__inj_I1_J,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( set_or5139330845457685135Than_a @ A @ B )
= ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_239_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_240_atLeastLessThan__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ( ( set_or5139330845457685135Than_a @ A @ B )
= ( set_or5139330845457685135Than_a @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_241_verit__comp__simplify1_I1_J,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_242_verit__comp__simplify1_I1_J,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_243_less__imp__neq,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_244_less__imp__neq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_245_order_Oasym,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order.asym
thf(fact_246_order_Oasym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order.asym
thf(fact_247_ord__eq__less__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_248_ord__eq__less__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_249_ord__less__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_250_ord__less__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_251_antisym__conv3,axiom,
! [Y: a,X: a] :
( ~ ( ord_less_a @ Y @ X )
=> ( ( ~ ( ord_less_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_252_linorder__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_a @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_253_dual__order_Oasym,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ~ ( ord_less_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_254_dual__order_Oasym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_255_dual__order_Oirrefl,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_256_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_257_linorder__less__wlog,axiom,
! [P2: a > a > $o,A: a,B: a] :
( ! [A5: a,B5: a] :
( ( ord_less_a @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: a] : ( P2 @ A5 @ A5 )
=> ( ! [A5: a,B5: a] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_258_order_Ostrict__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_259_order_Ostrict__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_260_top_Oextremum__strict,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ top_top_a_o @ A ) ).
% top.extremum_strict
thf(fact_261_top_Oextremum__strict,axiom,
! [A: set_set_a] :
~ ( ord_less_set_set_a @ top_top_set_set_a @ A ) ).
% top.extremum_strict
thf(fact_262_top_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ top_top_set_a @ A ) ).
% top.extremum_strict
thf(fact_263_top_Onot__eq__extremum,axiom,
! [A: a > $o] :
( ( A != top_top_a_o )
= ( ord_less_a_o @ A @ top_top_a_o ) ) ).
% top.not_eq_extremum
thf(fact_264_top_Onot__eq__extremum,axiom,
! [A: set_set_a] :
( ( A != top_top_set_set_a )
= ( ord_less_set_set_a @ A @ top_top_set_set_a ) ) ).
% top.not_eq_extremum
thf(fact_265_top_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != top_top_set_a )
= ( ord_less_set_a @ A @ top_top_set_a ) ) ).
% top.not_eq_extremum
thf(fact_266_not__less__iff__gr__or__eq,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_a @ X @ Y ) )
= ( ( ord_less_a @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_267_dual__order_Ostrict__trans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_268_dual__order_Ostrict__trans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_269_order_Ostrict__implies__not__eq,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_270_order_Ostrict__implies__not__eq,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_271_dual__order_Ostrict__implies__not__eq,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_272_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_273_ComplD,axiom,
! [C: set_a,A3: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A3 ) )
=> ~ ( member_set_a @ C @ A3 ) ) ).
% ComplD
thf(fact_274_ComplD,axiom,
! [C: a,A3: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) )
=> ~ ( member_a @ C @ A3 ) ) ).
% ComplD
thf(fact_275_UNIV__eq__I,axiom,
! [A3: set_set_a] :
( ! [X2: set_a] : ( member_set_a @ X2 @ A3 )
=> ( top_top_set_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_276_UNIV__eq__I,axiom,
! [A3: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A3 )
=> ( top_top_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_277_UNIV__witness,axiom,
? [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_278_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_279_double__complement,axiom,
! [A3: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_280_linorder__neqE,axiom,
! [X: a,Y: a] :
( ( X != Y )
=> ( ~ ( ord_less_a @ X @ Y )
=> ( ord_less_a @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_281_order__less__asym,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ~ ( ord_less_a @ Y @ X ) ) ).
% order_less_asym
thf(fact_282_order__less__asym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_asym
thf(fact_283_linorder__neq__iff,axiom,
! [X: a,Y: a] :
( ( X != Y )
= ( ( ord_less_a @ X @ Y )
| ( ord_less_a @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_284_order__less__asym_H,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_285_order__less__asym_H,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_286_order__less__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_a @ X @ Y )
=> ( ( ord_less_a @ Y @ Z )
=> ( ord_less_a @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_287_order__less__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_288_ord__eq__less__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_289_ord__eq__less__subst,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_290_ord__eq__less__subst,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_291_ord__eq__less__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_292_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_293_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_294_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_295_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_296_order__less__irrefl,axiom,
! [X: a] :
~ ( ord_less_a @ X @ X ) ).
% order_less_irrefl
thf(fact_297_order__less__irrefl,axiom,
! [X: set_a] :
~ ( ord_less_set_a @ X @ X ) ).
% order_less_irrefl
thf(fact_298_order__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_299_order__less__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_300_order__less__subst1,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_301_order__less__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_302_order__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_303_order__less__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_304_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_305_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_306_order__less__not__sym,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ~ ( ord_less_a @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_307_order__less__not__sym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_308_order__less__imp__triv,axiom,
! [X: a,Y: a,P2: $o] :
( ( ord_less_a @ X @ Y )
=> ( ( ord_less_a @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_309_order__less__imp__triv,axiom,
! [X: set_a,Y: set_a,P2: $o] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_310_linorder__less__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
| ( X = Y )
| ( ord_less_a @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_311_order__less__imp__not__eq,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_312_order__less__imp__not__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_313_order__less__imp__not__eq2,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_314_order__less__imp__not__eq2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_315_order__less__imp__not__less,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ~ ( ord_less_a @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_316_order__less__imp__not__less,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_317_atMost__eq__UNIV__iff,axiom,
! [X: a > $o] :
( ( ( set_ord_atMost_a_o @ X )
= top_top_set_a_o2 )
= ( X = top_top_a_o ) ) ).
% atMost_eq_UNIV_iff
thf(fact_318_atMost__eq__UNIV__iff,axiom,
! [X: set_set_a] :
( ( ( set_or4016371710855203973_set_a @ X )
= top_to4027821306633060462_set_a )
= ( X = top_top_set_set_a ) ) ).
% atMost_eq_UNIV_iff
thf(fact_319_atMost__eq__UNIV__iff,axiom,
! [X: set_a] :
( ( ( set_ord_atMost_set_a @ X )
= top_top_set_set_a )
= ( X = top_top_set_a ) ) ).
% atMost_eq_UNIV_iff
thf(fact_320_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A @ B ) @ ( set_or2348907005316661231_set_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_set_a @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_321_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $o > a,B: $o > a,C: $o > a,D: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_or8441445163928040022st_o_a @ A @ B ) @ ( set_or4510008498168808314an_o_a @ C @ D ) )
= ( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ C @ A )
& ( ord_less_o_a @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_322_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A @ B ) @ ( set_or5139330845457685135Than_a @ C @ D ) )
= ( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ A )
& ( ord_less_a @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_323_lessThan__strict__subset__iff,axiom,
! [M2: a,N: a] :
( ( ord_less_set_a @ ( set_ord_lessThan_a @ M2 ) @ ( set_ord_lessThan_a @ N ) )
= ( ord_less_a @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_324_top_Oextremum__uniqueI,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ top_top_a_o @ A )
=> ( A = top_top_a_o ) ) ).
% top.extremum_uniqueI
thf(fact_325_top_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A )
=> ( A = top_top_set_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_326_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_327_top_Oextremum__unique,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ top_top_a_o @ A )
= ( A = top_top_a_o ) ) ).
% top.extremum_unique
thf(fact_328_top_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A )
= ( A = top_top_set_set_a ) ) ).
% top.extremum_unique
thf(fact_329_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_330_top__greatest,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ A @ top_top_a_o ) ).
% top_greatest
thf(fact_331_top__greatest,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ top_top_set_set_a ) ).
% top_greatest
thf(fact_332_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_333_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_334_order__le__imp__less__or__eq,axiom,
! [X: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X @ Y )
=> ( ( ord_less_o_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_335_order__le__imp__less__or__eq,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_336_linorder__le__less__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_a @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_337_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_338_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_339_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_340_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_341_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > $o > a,C: $o > a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_342_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > $o > a,C: $o > a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_343_order__less__le__subst1,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_344_order__less__le__subst1,axiom,
! [A: $o > a,F: a > $o > a,B: a,C: a] :
( ( ord_less_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_345_order__less__le__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_346_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_347_order__less__le__subst1,axiom,
! [A: $o > a,F: set_a > $o > a,B: set_a,C: set_a] :
( ( ord_less_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_348_order__less__le__subst1,axiom,
! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_349_order__less__le__subst1,axiom,
! [A: set_a,F: ( $o > a ) > set_a,B: $o > a,C: $o > a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_350_order__less__le__subst1,axiom,
! [A: $o > a,F: ( $o > a ) > $o > a,B: $o > a,C: $o > a] :
( ( ord_less_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_351_order__less__le__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_352_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_353_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_o_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_354_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_355_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_356_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > $o > a,C: $o > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_o_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_357_order__le__less__subst2,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_358_order__le__less__subst2,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > set_a,C: set_a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_359_order__le__less__subst2,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_o_a @ ( F @ B ) @ C )
=> ( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_360_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_361_order__le__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_362_order__le__less__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_363_order__le__less__subst1,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_364_order__le__less__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_365_order__le__less__subst1,axiom,
! [A: $o > a,F: a > $o > a,B: a,C: a] :
( ( ord_less_eq_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_366_order__le__less__subst1,axiom,
! [A: $o > a,F: set_a > $o > a,B: set_a,C: set_a] :
( ( ord_less_eq_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_367_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_368_order__less__le__trans,axiom,
! [X: $o > a,Y: $o > a,Z: $o > a] :
( ( ord_less_o_a @ X @ Y )
=> ( ( ord_less_eq_o_a @ Y @ Z )
=> ( ord_less_o_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_369_order__less__le__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z )
=> ( ord_less_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_370_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_371_order__le__less__trans,axiom,
! [X: $o > a,Y: $o > a,Z: $o > a] :
( ( ord_less_eq_o_a @ X @ Y )
=> ( ( ord_less_o_a @ Y @ Z )
=> ( ord_less_o_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_372_order__le__less__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_a @ Y @ Z )
=> ( ord_less_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_373_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_374_order__neq__le__trans,axiom,
! [A: $o > a,B: $o > a] :
( ( A != B )
=> ( ( ord_less_eq_o_a @ A @ B )
=> ( ord_less_o_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_375_order__neq__le__trans,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_376_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_377_order__le__neq__trans,axiom,
! [A: $o > a,B: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_o_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_378_order__le__neq__trans,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_379_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_380_order__less__imp__le,axiom,
! [X: $o > a,Y: $o > a] :
( ( ord_less_o_a @ X @ Y )
=> ( ord_less_eq_o_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_381_order__less__imp__le,axiom,
! [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_382_linorder__not__less,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_a @ X @ Y ) )
= ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_not_less
thf(fact_383_linorder__not__le,axiom,
! [X: a,Y: a] :
( ( ~ ( ord_less_eq_a @ X @ Y ) )
= ( ord_less_a @ Y @ X ) ) ).
% linorder_not_le
thf(fact_384_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_385_order__less__le,axiom,
( ord_less_o_a
= ( ^ [X3: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_386_order__less__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_387_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_set_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_388_order__le__less,axiom,
( ord_less_eq_o_a
= ( ^ [X3: $o > a,Y4: $o > a] :
( ( ord_less_o_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_389_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X3: a,Y4: a] :
( ( ord_less_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_390_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_391_dual__order_Ostrict__implies__order,axiom,
! [B: $o > a,A: $o > a] :
( ( ord_less_o_a @ B @ A )
=> ( ord_less_eq_o_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_392_dual__order_Ostrict__implies__order,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_eq_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_393_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_394_order_Ostrict__implies__order,axiom,
! [A: $o > a,B: $o > a] :
( ( ord_less_o_a @ A @ B )
=> ( ord_less_eq_o_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_395_order_Ostrict__implies__order,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_eq_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_396_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_397_dual__order_Ostrict__iff__not,axiom,
( ord_less_o_a
= ( ^ [B4: $o > a,A4: $o > a] :
( ( ord_less_eq_o_a @ B4 @ A4 )
& ~ ( ord_less_eq_o_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_398_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_399_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_400_dual__order_Ostrict__trans2,axiom,
! [B: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ C @ B )
=> ( ord_less_o_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_401_dual__order_Ostrict__trans2,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_402_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_403_dual__order_Ostrict__trans1,axiom,
! [B: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_o_a @ C @ B )
=> ( ord_less_o_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_404_dual__order_Ostrict__trans1,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_405_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_406_dual__order_Ostrict__iff__order,axiom,
( ord_less_o_a
= ( ^ [B4: $o > a,A4: $o > a] :
( ( ord_less_eq_o_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_407_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_408_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_409_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_o_a
= ( ^ [B4: $o > a,A4: $o > a] :
( ( ord_less_o_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_410_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_411_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_412_order_Ostrict__iff__not,axiom,
( ord_less_o_a
= ( ^ [A4: $o > a,B4: $o > a] :
( ( ord_less_eq_o_a @ A4 @ B4 )
& ~ ( ord_less_eq_o_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_413_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_414_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_415_order_Ostrict__trans2,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_o_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_416_order_Ostrict__trans2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_417_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_418_order_Ostrict__trans1,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_o_a @ B @ C )
=> ( ord_less_o_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_419_order_Ostrict__trans1,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_420_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_421_order_Ostrict__iff__order,axiom,
( ord_less_o_a
= ( ^ [A4: $o > a,B4: $o > a] :
( ( ord_less_eq_o_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_422_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_423_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_424_order_Oorder__iff__strict,axiom,
( ord_less_eq_o_a
= ( ^ [A4: $o > a,B4: $o > a] :
( ( ord_less_o_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_425_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_426_not__le__imp__less,axiom,
! [Y: a,X: a] :
( ~ ( ord_less_eq_a @ Y @ X )
=> ( ord_less_a @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_427_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_428_less__le__not__le,axiom,
( ord_less_o_a
= ( ^ [X3: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X3 @ Y4 )
& ~ ( ord_less_eq_o_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_429_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
& ~ ( ord_less_eq_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_430_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_431_antisym__conv2,axiom,
! [X: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X @ Y )
=> ( ( ~ ( ord_less_o_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_432_antisym__conv2,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ~ ( ord_less_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_433_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_434_antisym__conv1,axiom,
! [X: $o > a,Y: $o > a] :
( ~ ( ord_less_o_a @ X @ Y )
=> ( ( ord_less_eq_o_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_435_antisym__conv1,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_436_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_437_nless__le,axiom,
! [A: $o > a,B: $o > a] :
( ( ~ ( ord_less_o_a @ A @ B ) )
= ( ~ ( ord_less_eq_o_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_438_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_439_leI,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% leI
thf(fact_440_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_441_leD,axiom,
! [Y: $o > a,X: $o > a] :
( ( ord_less_eq_o_a @ Y @ X )
=> ~ ( ord_less_o_a @ X @ Y ) ) ).
% leD
thf(fact_442_leD,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_a @ X @ Y ) ) ).
% leD
thf(fact_443_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_444_psubsetE,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_set_a @ A3 @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ).
% psubsetE
thf(fact_445_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
& ( A2 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_446_psubset__imp__subset,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_set_a @ A3 @ B2 )
=> ( ord_less_eq_set_a @ A3 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_447_psubset__subset__trans,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_448_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_449_subset__psubset__trans,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_450_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
| ( A2 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_451_less__fun__def,axiom,
( ord_less_o_a
= ( ^ [F2: $o > a,G: $o > a] :
( ( ord_less_eq_o_a @ F2 @ G )
& ~ ( ord_less_eq_o_a @ G @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_452_subset__UNIV,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ A3 @ top_top_set_set_a ) ).
% subset_UNIV
thf(fact_453_subset__UNIV,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_454_atLeastLessThan__subset__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ A @ B ) @ ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_eq_a @ B @ A )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_455_compl__less__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_set_a @ Y @ X ) ) ).
% compl_less_compl_iff
thf(fact_456_compl__le__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_457_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( uminus_uminus_set_a @ X )
= ( uminus_uminus_set_a @ Y ) )
= ( X = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_458_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ X ) )
= X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_459_psubsetD,axiom,
! [A3: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A3 @ B2 )
=> ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_460_psubsetD,axiom,
! [A3: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A3 @ B2 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_461_psubset__trans,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% psubset_trans
thf(fact_462_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_463_top__set__def,axiom,
( top_top_set_set_a
= ( collect_set_a @ top_top_set_a_o ) ) ).
% top_set_def
thf(fact_464_compl__mono,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% compl_mono
thf(fact_465_compl__le__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_466_compl__le__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_467_compl__less__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_less_swap1
thf(fact_468_compl__less__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_less_swap2
thf(fact_469_iso__tuple__UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_470_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_471_minf_I8_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ~ ( ord_less_eq_a @ T3 @ X4 ) ) ).
% minf(8)
thf(fact_472_minf_I6_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ord_less_eq_a @ X4 @ T3 ) ) ).
% minf(6)
thf(fact_473_pinf_I8_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ord_less_eq_a @ T3 @ X4 ) ) ).
% pinf(8)
thf(fact_474_pinf_I6_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ~ ( ord_less_eq_a @ X4 @ T3 ) ) ).
% pinf(6)
thf(fact_475_top_Oordering__top__axioms,axiom,
ordering_top_a_o @ ord_less_eq_a_o @ ord_less_a_o @ top_top_a_o ).
% top.ordering_top_axioms
thf(fact_476_top_Oordering__top__axioms,axiom,
orderi5875812994216768367_set_a @ ord_le3724670747650509150_set_a @ ord_less_set_set_a @ top_top_set_set_a ).
% top.ordering_top_axioms
thf(fact_477_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_478_top1I,axiom,
! [X: a] : ( top_top_a_o @ X ) ).
% top1I
thf(fact_479_pinf_I1_J,axiom,
! [P2: a > $o,P3: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X2: a] :
( ( ord_less_a @ Z4 @ X2 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: a] :
! [X2: a] :
( ( ord_less_a @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P3 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_480_pinf_I2_J,axiom,
! [P2: a > $o,P3: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X2: a] :
( ( ord_less_a @ Z4 @ X2 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: a] :
! [X2: a] :
( ( ord_less_a @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P3 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_481_pinf_I3_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(3)
thf(fact_482_pinf_I4_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(4)
thf(fact_483_pinf_I5_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ~ ( ord_less_a @ X4 @ T3 ) ) ).
% pinf(5)
thf(fact_484_pinf_I7_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ord_less_a @ T3 @ X4 ) ) ).
% pinf(7)
thf(fact_485_minf_I1_J,axiom,
! [P2: a > $o,P3: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z4 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P3 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_486_minf_I2_J,axiom,
! [P2: a > $o,P3: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z4 )
=> ( ( P2 @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P3 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_487_minf_I3_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( X4 != T3 ) ) ).
% minf(3)
thf(fact_488_minf_I4_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( X4 != T3 ) ) ).
% minf(4)
thf(fact_489_minf_I5_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ord_less_a @ X4 @ T3 ) ) ).
% minf(5)
thf(fact_490_minf_I7_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ~ ( ord_less_a @ T3 @ X4 ) ) ).
% minf(7)
thf(fact_491_top__empty__eq,axiom,
( top_top_set_a_o
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ) ) ).
% top_empty_eq
thf(fact_492_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_493_top__conj_I1_J,axiom,
! [X: a,P2: $o] :
( ( ( top_top_a_o @ X )
& P2 )
= P2 ) ).
% top_conj(1)
thf(fact_494_top__conj_I2_J,axiom,
! [P2: $o,X: a] :
( ( P2
& ( top_top_a_o @ X ) )
= P2 ) ).
% top_conj(2)
thf(fact_495_ivl__disj__un__one_I7_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ U ) @ ( set_or8632414552788122084Than_a @ U ) )
= ( set_ord_atLeast_a @ L ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_496_sup_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ B )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_497_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_498_sup_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_499_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_500_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_501_Un__iff,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B2 ) )
= ( ( member_set_a @ C @ A3 )
| ( member_set_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_502_Un__iff,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B2 ) )
= ( ( member_a @ C @ A3 )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_503_UnCI,axiom,
! [C: set_a,B2: set_set_a,A3: set_set_a] :
( ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ A3 ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_504_UnCI,axiom,
! [C: a,B2: set_a,A3: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A3 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_505_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_506_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_507_sup__top__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X )
= top_top_set_a ) ).
% sup_top_left
thf(fact_508_sup__top__left,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ top_top_a_o @ X )
= top_top_a_o ) ).
% sup_top_left
thf(fact_509_sup__top__left,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ top_top_set_set_a @ X )
= top_top_set_set_a ) ).
% sup_top_left
thf(fact_510_sup__top__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% sup_top_right
thf(fact_511_sup__top__right,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ X @ top_top_a_o )
= top_top_a_o ) ).
% sup_top_right
thf(fact_512_sup__top__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ top_top_set_set_a )
= top_top_set_set_a ) ).
% sup_top_right
thf(fact_513_boolean__algebra_Odisj__one__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X )
= top_top_set_a ) ).
% boolean_algebra.disj_one_left
thf(fact_514_boolean__algebra_Odisj__one__left,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ top_top_a_o @ X )
= top_top_a_o ) ).
% boolean_algebra.disj_one_left
thf(fact_515_boolean__algebra_Odisj__one__left,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ top_top_set_set_a @ X )
= top_top_set_set_a ) ).
% boolean_algebra.disj_one_left
thf(fact_516_boolean__algebra_Odisj__one__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% boolean_algebra.disj_one_right
thf(fact_517_boolean__algebra_Odisj__one__right,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ X @ top_top_a_o )
= top_top_a_o ) ).
% boolean_algebra.disj_one_right
thf(fact_518_boolean__algebra_Odisj__one__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ top_top_set_set_a )
= top_top_set_set_a ) ).
% boolean_algebra.disj_one_right
thf(fact_519_Un__subset__iff,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_a @ A3 @ C2 )
& ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_520_greaterThanLessThan__iff,axiom,
! [I: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I @ ( set_or6017932776736107018_set_a @ L @ U ) )
= ( ( ord_less_set_a @ L @ I )
& ( ord_less_set_a @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_521_greaterThanLessThan__iff,axiom,
! [I: a,L: a,U: a] :
( ( member_a @ I @ ( set_or5939364468397584554Than_a @ L @ U ) )
= ( ( ord_less_a @ L @ I )
& ( ord_less_a @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_522_sup__compl__top__left1,axiom,
! [X: a > $o,Y: a > $o] :
( ( sup_sup_a_o @ ( uminus_uminus_a_o @ X ) @ ( sup_sup_a_o @ X @ Y ) )
= top_top_a_o ) ).
% sup_compl_top_left1
thf(fact_523_sup__compl__top__left1,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ ( sup_sup_set_set_a @ X @ Y ) )
= top_top_set_set_a ) ).
% sup_compl_top_left1
thf(fact_524_sup__compl__top__left1,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ ( sup_sup_set_a @ X @ Y ) )
= top_top_set_a ) ).
% sup_compl_top_left1
thf(fact_525_sup__compl__top__left2,axiom,
! [X: a > $o,Y: a > $o] :
( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ ( uminus_uminus_a_o @ X ) @ Y ) )
= top_top_a_o ) ).
% sup_compl_top_left2
thf(fact_526_sup__compl__top__left2,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ Y ) )
= top_top_set_set_a ) ).
% sup_compl_top_left2
thf(fact_527_sup__compl__top__left2,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) )
= top_top_set_a ) ).
% sup_compl_top_left2
thf(fact_528_boolean__algebra_Odisj__cancel__left,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ ( uminus_uminus_a_o @ X ) @ X )
= top_top_a_o ) ).
% boolean_algebra.disj_cancel_left
thf(fact_529_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ X )
= top_top_set_set_a ) ).
% boolean_algebra.disj_cancel_left
thf(fact_530_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ X )
= top_top_set_a ) ).
% boolean_algebra.disj_cancel_left
thf(fact_531_boolean__algebra_Odisj__cancel__right,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ X @ ( uminus_uminus_a_o @ X ) )
= top_top_a_o ) ).
% boolean_algebra.disj_cancel_right
thf(fact_532_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( uminus6103902357914783669_set_a @ X ) )
= top_top_set_set_a ) ).
% boolean_algebra.disj_cancel_right
thf(fact_533_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ ( uminus_uminus_set_a @ X ) )
= top_top_set_a ) ).
% boolean_algebra.disj_cancel_right
thf(fact_534_greaterThanAtMost__iff,axiom,
! [I: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I @ ( set_or2503527069484367278_set_a @ L @ U ) )
= ( ( ord_less_set_a @ L @ I )
& ( ord_less_eq_set_a @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_535_greaterThanAtMost__iff,axiom,
! [I: $o > a,L: $o > a,U: $o > a] :
( ( member_o_a @ I @ ( set_or1794473594002085499st_o_a @ L @ U ) )
= ( ( ord_less_o_a @ L @ I )
& ( ord_less_eq_o_a @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_536_greaterThanAtMost__iff,axiom,
! [I: a,L: a,U: a] :
( ( member_a @ I @ ( set_or4472690218693186638Most_a @ L @ U ) )
= ( ( ord_less_a @ L @ I )
& ( ord_less_eq_a @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_537_Un__left__commute,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_538_Un__left__absorb,axiom,
! [A3: set_a,B2: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B2 ) )
= ( sup_sup_set_a @ A3 @ B2 ) ) ).
% Un_left_absorb
thf(fact_539_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A2: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A2 ) ) ) ).
% Un_commute
thf(fact_540_Un__absorb,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_541_Un__assoc,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ C2 )
= ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_542_ball__Un,axiom,
! [A3: set_a,B2: set_a,P2: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A3 @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( P2 @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_543_bex__Un,axiom,
! [A3: set_a,B2: set_a,P2: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A3 @ B2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A3 )
& ( P2 @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_544_UnI2,axiom,
! [C: set_a,B2: set_set_a,A3: set_set_a] :
( ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_545_UnI2,axiom,
! [C: a,B2: set_a,A3: set_a] :
( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_546_UnI1,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_547_UnI1,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ A3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_548_UnE,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A3 @ B2 ) )
=> ( ~ ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_549_UnE,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B2 ) )
=> ( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_550_ivl__disj__un__two_I6_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or4472690218693186638Most_a @ M2 @ U ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_551_ivl__disj__un__two__touch_I1_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_552_ivl__disj__un__two_I2_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or5939364468397584554Than_a @ M2 @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_553_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_554_sup_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_555_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_a,K: set_a,B: set_a,A: set_a] :
( ( B2
= ( sup_sup_set_a @ K @ B ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_556_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_a,K: set_a,A: set_a,B: set_a] :
( ( A3
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A3 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_557_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_558_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_559_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_560_sup_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.assoc
thf(fact_561_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_562_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_563_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_564_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_565_ivl__disj__un__two_I5_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_566_ivl__disj__un__two_I8_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or4472690218693186638Most_a @ M2 @ U ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_567_ivl__disj__un__one_I3_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or4472690218693186638Most_a @ L @ U ) )
= ( set_ord_atMost_a @ U ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_568_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_569_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_570_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_571_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_572_sup_Ocobounded2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded2
thf(fact_573_sup_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded1
thf(fact_574_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_575_sup_OboundedI,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_576_sup_OboundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_577_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_578_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_579_sup_Oabsorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_580_sup_Oabsorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_581_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_582_sup_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% sup.orderI
thf(fact_583_sup_OorderE,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( A
= ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.orderE
thf(fact_584_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_585_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_586_sup__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_587_sup_Omono,axiom,
! [C: set_a,A: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_588_le__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_589_le__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_590_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_591_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_592_le__supI,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_593_le__supE,axiom,
! [A: set_a,B: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B @ X ) ) ) ).
% le_supE
thf(fact_594_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_595_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_596_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_597_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ C @ A )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_598_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_599_sup_Ostrict__boundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_600_sup_Oabsorb4,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_601_sup_Oabsorb3,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_602_less__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ X @ B )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI2
thf(fact_603_less__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ X @ A )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI1
thf(fact_604_ivl__disj__un__one_I5_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ U ) @ ( set_or8632414552788122084Than_a @ U ) )
= ( set_or8632414552788122084Than_a @ L ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_605_Un__UNIV__right,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ top_top_set_a )
= top_top_set_a ) ).
% Un_UNIV_right
thf(fact_606_Un__UNIV__right,axiom,
! [A3: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ top_top_set_set_a )
= top_top_set_set_a ) ).
% Un_UNIV_right
thf(fact_607_Un__UNIV__left,axiom,
! [B2: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ B2 )
= top_top_set_a ) ).
% Un_UNIV_left
thf(fact_608_Un__UNIV__left,axiom,
! [B2: set_set_a] :
( ( sup_sup_set_set_a @ top_top_set_set_a @ B2 )
= top_top_set_set_a ) ).
% Un_UNIV_left
thf(fact_609_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B3: set_a] :
( ( sup_sup_set_a @ A2 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_610_subset__UnE,axiom,
! [C2: set_a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A3 @ B2 ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A3 )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B2 )
=> ( C2
!= ( sup_sup_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_611_Un__absorb2,axiom,
! [B2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( ( sup_sup_set_a @ A3 @ B2 )
= A3 ) ) ).
% Un_absorb2
thf(fact_612_Un__absorb1,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( sup_sup_set_a @ A3 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_613_Un__upper2,axiom,
! [B2: set_a,A3: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A3 @ B2 ) ) ).
% Un_upper2
thf(fact_614_Un__upper1,axiom,
! [A3: set_a,B2: set_a] : ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B2 ) ) ).
% Un_upper1
thf(fact_615_Un__least,axiom,
! [A3: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_616_Un__mono,axiom,
! [A3: set_a,C2: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_617_Ioc__inj,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( set_or4472690218693186638Most_a @ A @ B )
= ( set_or4472690218693186638Most_a @ C @ D ) )
= ( ( ( ord_less_eq_a @ B @ A )
& ( ord_less_eq_a @ D @ C ) )
| ( ( A = C )
& ( B = D ) ) ) ) ).
% Ioc_inj
thf(fact_618_ivl__disj__un__two__touch_I3_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_619_ivl__disj__un__two_I1_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_620_ivl__disj__un__one_I1_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or5939364468397584554Than_a @ L @ U ) )
= ( set_ord_lessThan_a @ U ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_621_ivl__disj__un__one_I6_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= ( set_or8632414552788122084Than_a @ L ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_622_sup__cancel__left1,axiom,
! [X: a > $o,A: a > $o,B: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ X @ A ) @ ( sup_sup_a_o @ ( uminus_uminus_a_o @ X ) @ B ) )
= top_top_a_o ) ).
% sup_cancel_left1
thf(fact_623_sup__cancel__left1,axiom,
! [X: set_set_a,A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X @ A ) @ ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ B ) )
= top_top_set_set_a ) ).
% sup_cancel_left1
thf(fact_624_sup__cancel__left1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ A ) @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ B ) )
= top_top_set_a ) ).
% sup_cancel_left1
thf(fact_625_sup__cancel__left2,axiom,
! [X: a > $o,A: a > $o,B: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ ( uminus_uminus_a_o @ X ) @ A ) @ ( sup_sup_a_o @ X @ B ) )
= top_top_a_o ) ).
% sup_cancel_left2
thf(fact_626_sup__cancel__left2,axiom,
! [X: set_set_a,A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ A ) @ ( sup_sup_set_set_a @ X @ B ) )
= top_top_set_set_a ) ).
% sup_cancel_left2
thf(fact_627_sup__cancel__left2,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ A ) @ ( sup_sup_set_a @ X @ B ) )
= top_top_set_a ) ).
% sup_cancel_left2
thf(fact_628_ivl__disj__un__two_I4_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or5939364468397584554Than_a @ M2 @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_629_ivl__disj__un__two__touch_I4_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_630_ivl__disj__un__two_I3_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_631_Compl__partition2,axiom,
! [A3: set_set_a] :
( ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ A3 ) @ A3 )
= top_top_set_set_a ) ).
% Compl_partition2
thf(fact_632_Compl__partition2,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ ( uminus_uminus_set_a @ A3 ) @ A3 )
= top_top_set_a ) ).
% Compl_partition2
thf(fact_633_Compl__partition,axiom,
! [A3: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ ( uminus6103902357914783669_set_a @ A3 ) )
= top_top_set_set_a ) ).
% Compl_partition
thf(fact_634_Compl__partition,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ ( uminus_uminus_set_a @ A3 ) )
= top_top_set_a ) ).
% Compl_partition
thf(fact_635_Ioc__subset__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ A @ B ) @ ( set_or4472690218693186638Most_a @ C @ D ) )
= ( ( ord_less_eq_a @ B @ A )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_636_sup__shunt,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( sup_sup_a_o @ X @ Y )
= top_top_a_o )
= ( ord_less_eq_a_o @ ( uminus_uminus_a_o @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_637_sup__shunt,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( sup_sup_set_set_a @ X @ Y )
= top_top_set_set_a )
= ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_638_sup__shunt,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= top_top_set_a )
= ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_639_ivl__disj__un__two_I7_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_640_ivl__disj__un__one_I2_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or5139330845457685135Than_a @ L @ U ) )
= ( set_ord_lessThan_a @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_641_ivl__disj__un__one_I8_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= ( set_ord_atLeast_a @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_642_ivl__disj__un__two__touch_I2_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_643_ivl__disj__un__one_I4_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or672772299803893939Most_a @ L @ U ) )
= ( set_ord_atMost_a @ U ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_644_image__eqI,axiom,
! [B: set_a,F: a > set_a,X: a,A3: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A3 )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_645_image__eqI,axiom,
! [B: a,F: set_a > a,X: set_a,A3: set_set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_set_a @ X @ A3 )
=> ( member_a @ B @ ( image_set_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_646_image__eqI,axiom,
! [B: set_a,F: set_a > set_a,X: set_a,A3: set_set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_set_a @ X @ A3 )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_647_image__eqI,axiom,
! [B: a,F: a > a,X: a,A3: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ B @ ( image_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_648_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_649_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_650_all__not__in__conv,axiom,
! [A3: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_651_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_652_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_653_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X3: a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_654_image__empty,axiom,
! [F: set_a > set_a] :
( ( image_set_a_set_a @ F @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_655_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_656_empty__is__image,axiom,
! [F: set_a > set_a,A3: set_set_a] :
( ( bot_bot_set_set_a
= ( image_set_a_set_a @ F @ A3 ) )
= ( A3 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_657_empty__is__image,axiom,
! [F: a > a,A3: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_658_image__is__empty,axiom,
! [F: set_a > set_a,A3: set_set_a] :
( ( ( image_set_a_set_a @ F @ A3 )
= bot_bot_set_set_a )
= ( A3 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_659_image__is__empty,axiom,
! [F: a > a,A3: set_a] :
( ( ( image_a_a @ F @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_660_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_661_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_662_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_663_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_664_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_665_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_666_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_667_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_668_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_669_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_670_ball__empty,axiom,
! [P2: a > $o,X4: a] :
( ( member_a @ X4 @ bot_bot_set_a )
=> ( P2 @ X4 ) ) ).
% ball_empty
thf(fact_671_Un__empty,axiom,
! [A3: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A3 @ B2 )
= bot_bot_set_a )
= ( ( A3 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_672_atLeast__empty__triv,axiom,
( ( set_or8362275514725411625_set_a @ bot_bot_set_a )
= top_top_set_set_a ) ).
% atLeast_empty_triv
thf(fact_673_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_a_o @ bot_bot_a_o )
= top_top_a_o ) ).
% boolean_algebra.compl_zero
thf(fact_674_boolean__algebra_Ocompl__zero,axiom,
( ( uminus6103902357914783669_set_a @ bot_bot_set_set_a )
= top_top_set_set_a ) ).
% boolean_algebra.compl_zero
thf(fact_675_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% boolean_algebra.compl_zero
thf(fact_676_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_a_o @ top_top_a_o )
= bot_bot_a_o ) ).
% boolean_algebra.compl_one
thf(fact_677_boolean__algebra_Ocompl__one,axiom,
( ( uminus6103902357914783669_set_a @ top_top_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.compl_one
thf(fact_678_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.compl_one
thf(fact_679_atLeastatMost__empty__iff2,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_set_a
= ( set_or6288561110385358355_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_680_atLeastatMost__empty__iff2,axiom,
! [A: $o > a,B: $o > a] :
( ( bot_bot_set_o_a
= ( set_or8441445163928040022st_o_a @ A @ B ) )
= ( ~ ( ord_less_eq_o_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_681_atLeastatMost__empty__iff2,axiom,
! [A: a,B: a] :
( ( bot_bot_set_a
= ( set_or672772299803893939Most_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_682_atLeastatMost__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( set_or6288561110385358355_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ~ ( ord_less_eq_set_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_683_atLeastatMost__empty__iff,axiom,
! [A: $o > a,B: $o > a] :
( ( ( set_or8441445163928040022st_o_a @ A @ B )
= bot_bot_set_o_a )
= ( ~ ( ord_less_eq_o_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_684_atLeastatMost__empty__iff,axiom,
! [A: a,B: a] :
( ( ( set_or672772299803893939Most_a @ A @ B )
= bot_bot_set_a )
= ( ~ ( ord_less_eq_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_685_atLeastatMost__empty,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( set_or6288561110385358355_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% atLeastatMost_empty
thf(fact_686_atLeastatMost__empty,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ( set_or672772299803893939Most_a @ A @ B )
= bot_bot_set_a ) ) ).
% atLeastatMost_empty
thf(fact_687_atLeastLessThan__empty,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( set_or2348907005316661231_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_688_atLeastLessThan__empty,axiom,
! [B: $o > a,A: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( set_or4510008498168808314an_o_a @ A @ B )
= bot_bot_set_o_a ) ) ).
% atLeastLessThan_empty
thf(fact_689_atLeastLessThan__empty,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( set_or5139330845457685135Than_a @ A @ B )
= bot_bot_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_690_atLeastLessThan__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( set_or2348907005316661231_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ~ ( ord_less_set_a @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_691_atLeastLessThan__empty__iff,axiom,
! [A: a,B: a] :
( ( ( set_or5139330845457685135Than_a @ A @ B )
= bot_bot_set_a )
= ( ~ ( ord_less_a @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_692_atLeastLessThan__empty__iff2,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_set_a
= ( set_or2348907005316661231_set_a @ A @ B ) )
= ( ~ ( ord_less_set_a @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_693_atLeastLessThan__empty__iff2,axiom,
! [A: a,B: a] :
( ( bot_bot_set_a
= ( set_or5139330845457685135Than_a @ A @ B ) )
= ( ~ ( ord_less_a @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_694_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_695_greaterThanAtMost__empty,axiom,
! [L: $o > a,K: $o > a] :
( ( ord_less_eq_o_a @ L @ K )
=> ( ( set_or1794473594002085499st_o_a @ K @ L )
= bot_bot_set_o_a ) ) ).
% greaterThanAtMost_empty
thf(fact_696_greaterThanAtMost__empty,axiom,
! [L: a,K: a] :
( ( ord_less_eq_a @ L @ K )
=> ( ( set_or4472690218693186638Most_a @ K @ L )
= bot_bot_set_a ) ) ).
% greaterThanAtMost_empty
thf(fact_697_greaterThanAtMost__empty__iff2,axiom,
! [K: set_a,L: set_a] :
( ( bot_bot_set_set_a
= ( set_or2503527069484367278_set_a @ K @ L ) )
= ( ~ ( ord_less_set_a @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_698_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_699_greaterThanAtMost__empty__iff,axiom,
! [K: set_a,L: set_a] :
( ( ( set_or2503527069484367278_set_a @ K @ L )
= bot_bot_set_set_a )
= ( ~ ( ord_less_set_a @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_700_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_701_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_702_greaterThanLessThan__empty,axiom,
! [L: $o > a,K: $o > a] :
( ( ord_less_eq_o_a @ L @ K )
=> ( ( set_or8643659207013295391an_o_a @ K @ L )
= bot_bot_set_o_a ) ) ).
% greaterThanLessThan_empty
thf(fact_703_greaterThanLessThan__empty,axiom,
! [L: a,K: a] :
( ( ord_less_eq_a @ L @ K )
=> ( ( set_or5939364468397584554Than_a @ K @ L )
= bot_bot_set_a ) ) ).
% greaterThanLessThan_empty
thf(fact_704_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_705_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_706_imageI,axiom,
! [X: a,A3: set_a,F: a > set_a] :
( ( member_a @ X @ A3 )
=> ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_707_imageI,axiom,
! [X: set_a,A3: set_set_a,F: set_a > a] :
( ( member_set_a @ X @ A3 )
=> ( member_a @ ( F @ X ) @ ( image_set_a_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_708_imageI,axiom,
! [X: set_a,A3: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A3 )
=> ( member_set_a @ ( F @ X ) @ ( image_set_a_set_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_709_imageI,axiom,
! [X: a,A3: set_a,F: a > a] :
( ( member_a @ X @ A3 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_710_equals0D,axiom,
! [A3: set_set_a,A: set_a] :
( ( A3 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A3 ) ) ).
% equals0D
thf(fact_711_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_712_equals0I,axiom,
! [A3: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_713_equals0I,axiom,
! [A3: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_714_image__iff,axiom,
! [Z: a,F: a > a,A3: set_a] :
( ( member_a @ Z @ ( image_a_a @ F @ A3 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_715_image__iff,axiom,
! [Z: set_a,F: set_a > set_a,A3: set_set_a] :
( ( member_set_a @ Z @ ( image_set_a_set_a @ F @ A3 ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_716_bex__imageD,axiom,
! [F: a > a,A3: set_a,P2: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A3 ) )
& ( P2 @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_717_bex__imageD,axiom,
! [F: set_a > set_a,A3: set_set_a,P2: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( image_set_a_set_a @ F @ A3 ) )
& ( P2 @ X4 ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_718_ex__in__conv,axiom,
! [A3: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_719_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_720_image__cong,axiom,
! [M3: set_a,N2: set_a,F: a > a,G2: a > a] :
( ( M3 = N2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( image_a_a @ F @ M3 )
= ( image_a_a @ G2 @ N2 ) ) ) ) ).
% image_cong
thf(fact_721_image__cong,axiom,
! [M3: set_set_a,N2: set_set_a,F: set_a > set_a,G2: set_a > set_a] :
( ( M3 = N2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( image_set_a_set_a @ F @ M3 )
= ( image_set_a_set_a @ G2 @ N2 ) ) ) ) ).
% image_cong
thf(fact_722_ball__imageD,axiom,
! [F: a > a,A3: set_a,P2: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F @ A3 ) )
=> ( P2 @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_723_ball__imageD,axiom,
! [F: set_a > set_a,A3: set_set_a,P2: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( image_set_a_set_a @ F @ A3 ) )
=> ( P2 @ X2 ) )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_724_rev__image__eqI,axiom,
! [X: a,A3: set_a,B: set_a,F: a > set_a] :
( ( member_a @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_725_rev__image__eqI,axiom,
! [X: set_a,A3: set_set_a,B: a,F: set_a > a] :
( ( member_set_a @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_set_a_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_726_rev__image__eqI,axiom,
! [X: set_a,A3: set_set_a,B: set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_727_rev__image__eqI,axiom,
! [X: a,A3: set_a,B: a,F: a > a] :
( ( member_a @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_728_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_729_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_730_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_731_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_732_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_733_rangeI,axiom,
! [F: a > a,X: a] : ( member_a @ ( F @ X ) @ ( image_a_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_734_rangeI,axiom,
! [F: a > set_a,X: a] : ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_735_rangeI,axiom,
! [F: set_a > a,X: set_a] : ( member_a @ ( F @ X ) @ ( image_set_a_a @ F @ top_top_set_set_a ) ) ).
% rangeI
thf(fact_736_rangeI,axiom,
! [F: set_a > set_a,X: set_a] : ( member_set_a @ ( F @ X ) @ ( image_set_a_set_a @ F @ top_top_set_set_a ) ) ).
% rangeI
thf(fact_737_range__eqI,axiom,
! [B: a,F: a > a,X: a] :
( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_738_range__eqI,axiom,
! [B: set_a,F: a > set_a,X: a] :
( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_739_range__eqI,axiom,
! [B: a,F: set_a > a,X: set_a] :
( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_set_a_a @ F @ top_top_set_set_a ) ) ) ).
% range_eqI
thf(fact_740_range__eqI,axiom,
! [B: set_a,F: set_a > set_a,X: set_a] :
( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ top_top_set_set_a ) ) ) ).
% range_eqI
thf(fact_741_image__mono,axiom,
! [A3: set_set_a,B2: set_set_a,F: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A3 ) @ ( image_set_a_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_742_image__mono,axiom,
! [A3: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_743_image__subsetI,axiom,
! [A3: set_a,F: a > set_a,B2: set_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_set_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A3 ) @ B2 ) ) ).
% image_subsetI
thf(fact_744_image__subsetI,axiom,
! [A3: set_set_a,F: set_a > set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( member_set_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A3 ) @ B2 ) ) ).
% image_subsetI
thf(fact_745_image__subsetI,axiom,
! [A3: set_set_a,F: set_a > a,B2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A3 ) @ B2 ) ) ).
% image_subsetI
thf(fact_746_image__subsetI,axiom,
! [A3: set_a,F: a > a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ B2 ) ) ).
% image_subsetI
thf(fact_747_subset__imageE,axiom,
! [B2: set_set_a,F: set_a > set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_set_a_set_a @ F @ A3 ) )
=> ~ ! [C5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C5 @ A3 )
=> ( B2
!= ( image_set_a_set_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_748_subset__imageE,axiom,
! [B2: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A3 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A3 )
=> ( B2
!= ( image_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_749_image__subset__iff,axiom,
! [F: set_a > set_a,A3: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A3 ) @ B2 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( member_set_a @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_750_image__subset__iff,axiom,
! [F: a > a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ B2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_751_subset__image__iff,axiom,
! [B2: set_set_a,F: set_a > set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_set_a_set_a @ F @ A3 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A3 )
& ( B2
= ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_752_subset__image__iff,axiom,
! [B2: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B2
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_753_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_754_image__Un,axiom,
! [F: set_a > set_a,A3: set_set_a,B2: set_set_a] :
( ( image_set_a_set_a @ F @ ( sup_sup_set_set_a @ A3 @ B2 ) )
= ( sup_sup_set_set_a @ ( image_set_a_set_a @ F @ A3 ) @ ( image_set_a_set_a @ F @ B2 ) ) ) ).
% image_Un
thf(fact_755_image__Un,axiom,
! [F: a > a,A3: set_a,B2: set_a] :
( ( image_a_a @ F @ ( sup_sup_set_a @ A3 @ B2 ) )
= ( sup_sup_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_Un
thf(fact_756_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_757_empty__not__UNIV,axiom,
bot_bot_set_set_a != top_top_set_set_a ).
% empty_not_UNIV
thf(fact_758_Un__empty__left,axiom,
! [B2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_759_Un__empty__right,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Un_empty_right
thf(fact_760_not__psubset__empty,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_761_not__empty__eq__Iic__eq__empty,axiom,
! [H: a] :
( bot_bot_set_a
!= ( set_ord_atMost_a @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_762_pairwise__imageI,axiom,
! [A3: set_set_a,F: set_a > set_a,P2: set_a > set_a > $o] :
( ! [X2: set_a,Y3: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( member_set_a @ Y3 @ A3 )
=> ( ( X2 != Y3 )
=> ( ( ( F @ X2 )
!= ( F @ Y3 ) )
=> ( P2 @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) )
=> ( pairwise_set_a @ P2 @ ( image_set_a_set_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_763_pairwise__imageI,axiom,
! [A3: set_a,F: a > a,P2: a > a > $o] :
( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ A3 )
=> ( ( member_a @ Y3 @ A3 )
=> ( ( X2 != Y3 )
=> ( ( ( F @ X2 )
!= ( F @ Y3 ) )
=> ( P2 @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) )
=> ( pairwise_a @ P2 @ ( image_a_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_764_pairwise__imageI,axiom,
! [A3: set_set_a,F: set_a > a,P2: a > a > $o] :
( ! [X2: set_a,Y3: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( member_set_a @ Y3 @ A3 )
=> ( ( X2 != Y3 )
=> ( ( ( F @ X2 )
!= ( F @ Y3 ) )
=> ( P2 @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) )
=> ( pairwise_a @ P2 @ ( image_set_a_a @ F @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_765_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_766_pairwise__empty,axiom,
! [P2: a > a > $o] : ( pairwise_a @ P2 @ bot_bot_set_a ) ).
% pairwise_empty
thf(fact_767_range__subsetD,axiom,
! [F: a > set_a,B2: set_set_a,I: a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ top_top_set_a ) @ B2 )
=> ( member_set_a @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_768_range__subsetD,axiom,
! [F: set_a > set_a,B2: set_set_a,I: set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ top_top_set_set_a ) @ B2 )
=> ( member_set_a @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_769_range__subsetD,axiom,
! [F: a > a,B2: set_a,I: a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ top_top_set_a ) @ B2 )
=> ( member_a @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_770_range__subsetD,axiom,
! [F: set_a > a,B2: set_a,I: set_a] :
( ( ord_less_eq_set_a @ ( image_set_a_a @ F @ top_top_set_set_a ) @ B2 )
=> ( member_a @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_771_atLeast__eq__UNIV__iff,axiom,
! [X: set_a] :
( ( ( set_or8362275514725411625_set_a @ X )
= top_top_set_set_a )
= ( X = bot_bot_set_a ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_772_Compl__UNIV__eq,axiom,
( ( uminus6103902357914783669_set_a @ top_top_set_set_a )
= bot_bot_set_set_a ) ).
% Compl_UNIV_eq
thf(fact_773_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% Compl_UNIV_eq
thf(fact_774_Compl__empty__eq,axiom,
( ( uminus6103902357914783669_set_a @ bot_bot_set_set_a )
= top_top_set_set_a ) ).
% Compl_empty_eq
thf(fact_775_Compl__empty__eq,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% Compl_empty_eq
thf(fact_776_subset__Compl__self__eq,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_777_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( set_or497184483940929162st_a_o @ X @ Y )
= top_top_set_a_o2 )
= ( ( X = bot_bot_a_o )
& ( Y = top_top_a_o ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_778_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( set_or4761464488706262899_set_a @ X @ Y )
= top_to4027821306633060462_set_a )
= ( ( X = bot_bot_set_set_a )
& ( Y = top_top_set_set_a ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_779_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( set_or6288561110385358355_set_a @ X @ Y )
= top_top_set_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = top_top_set_a ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_780_surj__Compl__image__subset,axiom,
! [F: set_a > set_a,A3: set_set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ ( image_set_a_set_a @ F @ A3 ) ) @ ( image_set_a_set_a @ F @ ( uminus6103902357914783669_set_a @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_781_surj__Compl__image__subset,axiom,
! [F: a > set_a,A3: set_a] :
( ( ( image_a_set_a @ F @ top_top_set_a )
= top_top_set_set_a )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ ( image_a_set_a @ F @ A3 ) ) @ ( image_a_set_a @ F @ ( uminus_uminus_set_a @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_782_surj__Compl__image__subset,axiom,
! [F: set_a > a,A3: set_set_a] :
( ( ( image_set_a_a @ F @ top_top_set_set_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_set_a_a @ F @ A3 ) ) @ ( image_set_a_a @ F @ ( uminus6103902357914783669_set_a @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_783_surj__Compl__image__subset,axiom,
! [F: a > a,A3: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_a_a @ F @ A3 ) ) @ ( image_a_a @ F @ ( uminus_uminus_set_a @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_784_subset__emptyI,axiom,
! [A3: set_set_a] :
( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A3 )
=> ( ord_le3724670747650509150_set_a @ A3 @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_785_subset__emptyI,axiom,
! [A3: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_786_all__subset__image,axiom,
! [F: set_a > set_a,A3: set_set_a,P2: set_set_a > $o] :
( ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_set_a_set_a @ F @ A3 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
=> ( P2 @ ( image_set_a_set_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_787_all__subset__image,axiom,
! [F: a > a,A3: set_a,P2: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A3 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( P2 @ ( image_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_788_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_789_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y4: a] :
? [X3: a] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_790_surj__def,axiom,
! [F: a > set_a] :
( ( ( image_a_set_a @ F @ top_top_set_a )
= top_top_set_set_a )
= ( ! [Y4: set_a] :
? [X3: a] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_791_surj__def,axiom,
! [F: set_a > a] :
( ( ( image_set_a_a @ F @ top_top_set_set_a )
= top_top_set_a )
= ( ! [Y4: a] :
? [X3: set_a] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_792_surj__def,axiom,
! [F: set_a > set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
= ( ! [Y4: set_a] :
? [X3: set_a] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_793_surjI,axiom,
! [G2: a > a,F: a > a] :
( ! [X2: a] :
( ( G2 @ ( F @ X2 ) )
= X2 )
=> ( ( image_a_a @ G2 @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_794_surjI,axiom,
! [G2: a > set_a,F: set_a > a] :
( ! [X2: set_a] :
( ( G2 @ ( F @ X2 ) )
= X2 )
=> ( ( image_a_set_a @ G2 @ top_top_set_a )
= top_top_set_set_a ) ) ).
% surjI
thf(fact_795_surjI,axiom,
! [G2: set_a > a,F: a > set_a] :
( ! [X2: a] :
( ( G2 @ ( F @ X2 ) )
= X2 )
=> ( ( image_set_a_a @ G2 @ top_top_set_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_796_surjI,axiom,
! [G2: set_a > set_a,F: set_a > set_a] :
( ! [X2: set_a] :
( ( G2 @ ( F @ X2 ) )
= X2 )
=> ( ( image_set_a_set_a @ G2 @ top_top_set_set_a )
= top_top_set_set_a ) ) ).
% surjI
thf(fact_797_surjE,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X2: a] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_798_surjE,axiom,
! [F: a > set_a,Y: set_a] :
( ( ( image_a_set_a @ F @ top_top_set_a )
= top_top_set_set_a )
=> ~ ! [X2: a] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_799_surjE,axiom,
! [F: set_a > a,Y: a] :
( ( ( image_set_a_a @ F @ top_top_set_set_a )
= top_top_set_a )
=> ~ ! [X2: set_a] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_800_surjE,axiom,
! [F: set_a > set_a,Y: set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
=> ~ ! [X2: set_a] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_801_surjD,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ? [X2: a] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_802_surjD,axiom,
! [F: a > set_a,Y: set_a] :
( ( ( image_a_set_a @ F @ top_top_set_a )
= top_top_set_set_a )
=> ? [X2: a] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_803_surjD,axiom,
! [F: set_a > a,Y: a] :
( ( ( image_set_a_a @ F @ top_top_set_set_a )
= top_top_set_a )
=> ? [X2: set_a] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_804_surjD,axiom,
! [F: set_a > set_a,Y: set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
=> ? [X2: set_a] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_805_ivl__disj__un__singleton_I4_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ U ) @ ( insert_a @ U @ bot_bot_set_a ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_806_ivl__disj__un__singleton_I3_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( insert_a @ L @ bot_bot_set_a ) @ ( set_or5939364468397584554Than_a @ L @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_807_the__elem__image__unique,axiom,
! [A3: set_set_a,F: set_a > set_a,X: set_a] :
( ( A3 != bot_bot_set_set_a )
=> ( ! [Y3: set_a] :
( ( member_set_a @ Y3 @ A3 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_set_a @ ( image_set_a_set_a @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_808_the__elem__image__unique,axiom,
! [A3: set_a,F: a > a,X: a] :
( ( A3 != bot_bot_set_a )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ A3 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_a @ ( image_a_a @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_809_ivl__disj__un__singleton_I5_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( insert_a @ L @ bot_bot_set_a ) @ ( set_or4472690218693186638Most_a @ L @ U ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_810_insertCI,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B2 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_811_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_812_insert__iff,axiom,
! [A: set_a,B: set_a,A3: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
= ( ( A = B )
| ( member_set_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_813_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_814_insert__absorb2,axiom,
! [X: a,A3: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A3 ) )
= ( insert_a @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_815_insert__image,axiom,
! [X: a,A3: set_a,F: a > a] :
( ( member_a @ X @ A3 )
=> ( ( insert_a @ ( F @ X ) @ ( image_a_a @ F @ A3 ) )
= ( image_a_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_816_insert__image,axiom,
! [X: set_a,A3: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A3 )
=> ( ( insert_set_a @ ( F @ X ) @ ( image_set_a_set_a @ F @ A3 ) )
= ( image_set_a_set_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_817_insert__image,axiom,
! [X: set_a,A3: set_set_a,F: set_a > a] :
( ( member_set_a @ X @ A3 )
=> ( ( insert_a @ ( F @ X ) @ ( image_set_a_a @ F @ A3 ) )
= ( image_set_a_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_818_image__insert,axiom,
! [F: set_a > set_a,A: set_a,B2: set_set_a] :
( ( image_set_a_set_a @ F @ ( insert_set_a @ A @ B2 ) )
= ( insert_set_a @ ( F @ A ) @ ( image_set_a_set_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_819_image__insert,axiom,
! [F: a > a,A: a,B2: set_a] :
( ( image_a_a @ F @ ( insert_a @ A @ B2 ) )
= ( insert_a @ ( F @ A ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_820_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_821_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_822_insert__subset,axiom,
! [X: set_a,A3: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A3 ) @ B2 )
= ( ( member_set_a @ X @ B2 )
& ( ord_le3724670747650509150_set_a @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_823_insert__subset,axiom,
! [X: a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_824_Un__insert__right,axiom,
! [A3: set_a,A: a,B2: set_a] :
( ( sup_sup_set_a @ A3 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A3 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_825_Un__insert__left,axiom,
! [A: a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_826_singleton__insert__inj__eq_H,axiom,
! [A: a,A3: set_a,B: a] :
( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_827_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A3: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_828_atLeastAtMost__singleton__iff,axiom,
! [A: a,B: a,C: a] :
( ( ( set_or672772299803893939Most_a @ A @ B )
= ( insert_a @ C @ bot_bot_set_a ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_829_atLeastAtMost__singleton,axiom,
! [A: a] :
( ( set_or672772299803893939Most_a @ A @ A )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% atLeastAtMost_singleton
thf(fact_830_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_831_subset__Compl__singleton,axiom,
! [A3: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( uminus6103902357914783669_set_a @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) )
= ( ~ ( member_set_a @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_832_subset__Compl__singleton,axiom,
! [A3: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_833_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_834_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_835_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_836_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_837_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_838_insert__not__empty,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_839_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_840_insert__subsetI,axiom,
! [X: set_a,A3: set_set_a,X6: set_set_a] :
( ( member_set_a @ X @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ X6 @ A3 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ X6 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_841_insert__subsetI,axiom,
! [X: a,A3: set_a,X6: set_a] :
( ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ X6 @ A3 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X6 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_842_pairwise__insert,axiom,
! [R3: set_a > set_a > $o,X: set_a,S3: set_set_a] :
( ( pairwise_set_a @ R3 @ ( insert_set_a @ X @ S3 ) )
= ( ! [Y4: set_a] :
( ( ( member_set_a @ Y4 @ S3 )
& ( Y4 != X ) )
=> ( ( R3 @ X @ Y4 )
& ( R3 @ Y4 @ X ) ) )
& ( pairwise_set_a @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_843_pairwise__insert,axiom,
! [R3: a > a > $o,X: a,S3: set_a] :
( ( pairwise_a @ R3 @ ( insert_a @ X @ S3 ) )
= ( ! [Y4: a] :
( ( ( member_a @ Y4 @ S3 )
& ( Y4 != X ) )
=> ( ( R3 @ X @ Y4 )
& ( R3 @ Y4 @ X ) ) )
& ( pairwise_a @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_844_insertE,axiom,
! [A: set_a,B: set_a,A3: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_845_insertE,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_846_insertI1,axiom,
! [A: set_a,B2: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B2 ) ) ).
% insertI1
thf(fact_847_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_848_insertI2,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( member_set_a @ A @ B2 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_849_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_850_Set_Oset__insert,axiom,
! [X: set_a,A3: set_set_a] :
( ( member_set_a @ X @ A3 )
=> ~ ! [B8: set_set_a] :
( ( A3
= ( insert_set_a @ X @ B8 ) )
=> ( member_set_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_851_Set_Oset__insert,axiom,
! [X: a,A3: set_a] :
( ( member_a @ X @ A3 )
=> ~ ! [B8: set_a] :
( ( A3
= ( insert_a @ X @ B8 ) )
=> ( member_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_852_insert__ident,axiom,
! [X: set_a,A3: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A3 )
=> ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( insert_set_a @ X @ A3 )
= ( insert_set_a @ X @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_853_insert__ident,axiom,
! [X: a,A3: set_a,B2: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A3 )
= ( insert_a @ X @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_854_insert__absorb,axiom,
! [A: set_a,A3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ( ( insert_set_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_855_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_856_insert__eq__iff,axiom,
! [A: set_a,A3: set_set_a,B: set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ A3 )
=> ( ~ ( member_set_a @ B @ B2 )
=> ( ( ( insert_set_a @ A @ A3 )
= ( insert_set_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A3 = B2 ) )
& ( ( A != B )
=> ? [C6: set_set_a] :
( ( A3
= ( insert_set_a @ B @ C6 ) )
& ~ ( member_set_a @ B @ C6 )
& ( B2
= ( insert_set_a @ A @ C6 ) )
& ~ ( member_set_a @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_857_insert__eq__iff,axiom,
! [A: a,A3: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A3 = B2 ) )
& ( ( A != B )
=> ? [C6: set_a] :
( ( A3
= ( insert_a @ B @ C6 ) )
& ~ ( member_a @ B @ C6 )
& ( B2
= ( insert_a @ A @ C6 ) )
& ~ ( member_a @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_858_insert__commute,axiom,
! [X: a,Y: a,A3: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A3 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_859_mk__disjoint__insert,axiom,
! [A: set_a,A3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ? [B8: set_set_a] :
( ( A3
= ( insert_set_a @ A @ B8 ) )
& ~ ( member_set_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_860_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B8: set_a] :
( ( A3
= ( insert_a @ A @ B8 ) )
& ~ ( member_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_861_subset__insertI2,axiom,
! [A3: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_862_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_863_subset__insert,axiom,
! [X: set_a,A3: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ X @ B2 ) )
= ( ord_le3724670747650509150_set_a @ A3 @ B2 ) ) ) ).
% subset_insert
thf(fact_864_subset__insert,axiom,
! [X: a,A3: set_a,B2: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A3 @ B2 ) ) ) ).
% subset_insert
thf(fact_865_insert__mono,axiom,
! [C2: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_866_insert__UNIV,axiom,
! [X: a] :
( ( insert_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_867_insert__UNIV,axiom,
! [X: set_a] :
( ( insert_set_a @ X @ top_top_set_set_a )
= top_top_set_set_a ) ).
% insert_UNIV
thf(fact_868_subset__singleton__iff,axiom,
! [X6: set_a,A: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_869_subset__singletonD,axiom,
! [A3: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A3 = bot_bot_set_a )
| ( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_870_singleton__Un__iff,axiom,
! [X: a,A3: set_a,B2: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A3 @ B2 ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_871_Un__singleton__iff,axiom,
! [A3: set_a,B2: set_a,X: a] :
( ( ( sup_sup_set_a @ A3 @ B2 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_872_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_873_atLeastAtMost__singleton_H,axiom,
! [A: a,B: a] :
( ( A = B )
=> ( ( set_or672772299803893939Most_a @ A @ B )
= ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_874_pairwise__singleton,axiom,
! [P2: a > a > $o,A3: a] : ( pairwise_a @ P2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% pairwise_singleton
thf(fact_875_range__eq__singletonD,axiom,
! [F: a > a,A: a,X: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= ( insert_a @ A @ bot_bot_set_a ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_876_range__eq__singletonD,axiom,
! [F: set_a > set_a,A: set_a,X: set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_877_range__eq__singletonD,axiom,
! [F: set_a > a,A: a,X: set_a] :
( ( ( image_set_a_a @ F @ top_top_set_set_a )
= ( insert_a @ A @ bot_bot_set_a ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_878_ivl__disj__un__singleton_I2_J,axiom,
! [U: a] :
( ( sup_sup_set_a @ ( set_ord_lessThan_a @ U ) @ ( insert_a @ U @ bot_bot_set_a ) )
= ( set_ord_atMost_a @ U ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_879_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_880_ivl__disj__un__singleton_I6_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ U ) @ ( insert_a @ U @ bot_bot_set_a ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_881_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A2: set_a] :
( A2
= ( insert_a @ ( the_elem_a @ A2 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_882_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_883_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A2: set_a] : ( A2 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_884_is__singletonE,axiom,
! [A3: set_a] :
( ( is_singleton_a @ A3 )
=> ~ ! [X2: a] :
( A3
!= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_885_is__singletonI_H,axiom,
! [A3: set_set_a] :
( ( A3 != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y3: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( member_set_a @ Y3 @ A3 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_set_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_886_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ A3 )
=> ( ( member_a @ Y3 @ A3 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_887_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A2: set_a] :
? [X3: a] :
( A2
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_888_Inf__atMostLessThan,axiom,
! [X: a > $o] :
( ( ord_less_a_o @ top_top_a_o @ X )
=> ( ( complete_Inf_Inf_a_o @ ( set_ord_lessThan_a_o @ X ) )
= bot_bot_a_o ) ) ).
% Inf_atMostLessThan
thf(fact_889_Inf__atMostLessThan,axiom,
! [X: set_set_a] :
( ( ord_less_set_set_a @ top_top_set_set_a @ X )
=> ( ( comple9105089376463352645_set_a @ ( set_or5369375139905502561_set_a @ X ) )
= bot_bot_set_set_a ) ) ).
% Inf_atMostLessThan
thf(fact_890_Inf__atMostLessThan,axiom,
! [X: set_a] :
( ( ord_less_set_a @ top_top_set_a @ X )
=> ( ( comple6135023378680113637_set_a @ ( set_or5421148953861284865_set_a @ X ) )
= bot_bot_set_a ) ) ).
% Inf_atMostLessThan
thf(fact_891_mono__image__least,axiom,
! [F: set_a > set_a,M2: set_a,N: set_a,M4: set_a,N3: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ( image_set_a_set_a @ F @ ( set_or2348907005316661231_set_a @ M2 @ N ) )
= ( set_or2348907005316661231_set_a @ M4 @ N3 ) )
=> ( ( ord_less_set_a @ M2 @ N )
=> ( ( F @ M2 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_892_mono__image__least,axiom,
! [F: set_a > $o > a,M2: set_a,N: set_a,M4: $o > a,N3: $o > a] :
( ( monoto9194799565806279554_a_o_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_o_a @ F )
=> ( ( ( image_set_a_o_a @ F @ ( set_or2348907005316661231_set_a @ M2 @ N ) )
= ( set_or4510008498168808314an_o_a @ M4 @ N3 ) )
=> ( ( ord_less_set_a @ M2 @ N )
=> ( ( F @ M2 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_893_mono__image__least,axiom,
! [F: ( $o > a ) > set_a,M2: $o > a,N: $o > a,M4: set_a,N3: set_a] :
( ( monoto888385181794500650_set_a @ top_top_set_o_a @ ord_less_eq_o_a @ ord_less_eq_set_a @ F )
=> ( ( ( image_o_a_set_a @ F @ ( set_or4510008498168808314an_o_a @ M2 @ N ) )
= ( set_or2348907005316661231_set_a @ M4 @ N3 ) )
=> ( ( ord_less_o_a @ M2 @ N )
=> ( ( F @ M2 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_894_mono__image__least,axiom,
! [F: ( $o > a ) > $o > a,M2: $o > a,N: $o > a,M4: $o > a,N3: $o > a] :
( ( monotone_on_o_a_o_a @ top_top_set_o_a @ ord_less_eq_o_a @ ord_less_eq_o_a @ F )
=> ( ( ( image_o_a_o_a @ F @ ( set_or4510008498168808314an_o_a @ M2 @ N ) )
= ( set_or4510008498168808314an_o_a @ M4 @ N3 ) )
=> ( ( ord_less_o_a @ M2 @ N )
=> ( ( F @ M2 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_895_mono__image__least,axiom,
! [F: set_a > a,M2: set_a,N: set_a,M4: a,N3: a] :
( ( monotone_on_set_a_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_a @ F )
=> ( ( ( image_set_a_a @ F @ ( set_or2348907005316661231_set_a @ M2 @ N ) )
= ( set_or5139330845457685135Than_a @ M4 @ N3 ) )
=> ( ( ord_less_set_a @ M2 @ N )
=> ( ( F @ M2 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_896_mono__image__least,axiom,
! [F: ( $o > a ) > a,M2: $o > a,N: $o > a,M4: a,N3: a] :
( ( monotone_on_o_a_a @ top_top_set_o_a @ ord_less_eq_o_a @ ord_less_eq_a @ F )
=> ( ( ( image_o_a_a @ F @ ( set_or4510008498168808314an_o_a @ M2 @ N ) )
= ( set_or5139330845457685135Than_a @ M4 @ N3 ) )
=> ( ( ord_less_o_a @ M2 @ N )
=> ( ( F @ M2 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_897_mono__image__least,axiom,
! [F: a > set_a,M2: a,N: a,M4: set_a,N3: set_a] :
( ( monotone_on_a_set_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_set_a @ F )
=> ( ( ( image_a_set_a @ F @ ( set_or5139330845457685135Than_a @ M2 @ N ) )
= ( set_or2348907005316661231_set_a @ M4 @ N3 ) )
=> ( ( ord_less_a @ M2 @ N )
=> ( ( F @ M2 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_898_mono__image__least,axiom,
! [F: a > $o > a,M2: a,N: a,M4: $o > a,N3: $o > a] :
( ( monotone_on_a_o_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_o_a @ F )
=> ( ( ( image_a_o_a @ F @ ( set_or5139330845457685135Than_a @ M2 @ N ) )
= ( set_or4510008498168808314an_o_a @ M4 @ N3 ) )
=> ( ( ord_less_a @ M2 @ N )
=> ( ( F @ M2 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_899_mono__image__least,axiom,
! [F: a > a,M2: a,N: a,M4: a,N3: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ( image_a_a @ F @ ( set_or5139330845457685135Than_a @ M2 @ N ) )
= ( set_or5139330845457685135Than_a @ M4 @ N3 ) )
=> ( ( ord_less_a @ M2 @ N )
=> ( ( F @ M2 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_900_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_901_atLeastAtMost__diff__ends,axiom,
! [A: a,B: a] :
( ( minus_minus_set_a @ ( set_or672772299803893939Most_a @ A @ B ) @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( set_or5939364468397584554Than_a @ A @ B ) ) ).
% atLeastAtMost_diff_ends
thf(fact_902_Inter__UNIV__conv_I2_J,axiom,
! [A3: set_set_a] :
( ( top_top_set_a
= ( comple6135023378680113637_set_a @ A3 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( X3 = top_top_set_a ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_903_Inter__UNIV__conv_I2_J,axiom,
! [A3: set_set_set_a] :
( ( top_top_set_set_a
= ( comple9105089376463352645_set_a @ A3 ) )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A3 )
=> ( X3 = top_top_set_set_a ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_904_Inter__UNIV__conv_I1_J,axiom,
! [A3: set_set_a] :
( ( ( comple6135023378680113637_set_a @ A3 )
= top_top_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( X3 = top_top_set_a ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_905_Inter__UNIV__conv_I1_J,axiom,
! [A3: set_set_set_a] :
( ( ( comple9105089376463352645_set_a @ A3 )
= top_top_set_set_a )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A3 )
=> ( X3 = top_top_set_set_a ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_906_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_907_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_908_inf_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_909_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_910_inf_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ B )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_911_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_912_IntI,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A3 )
=> ( ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_913_IntI,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ A3 )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_914_Int__iff,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B2 ) )
= ( ( member_set_a @ C @ A3 )
& ( member_set_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_915_Int__iff,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B2 ) )
= ( ( member_a @ C @ A3 )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_916_DiffI,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A3 )
=> ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_917_DiffI,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ A3 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_918_Diff__iff,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B2 ) )
= ( ( member_set_a @ C @ A3 )
& ~ ( member_set_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_919_Diff__iff,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B2 ) )
= ( ( member_a @ C @ A3 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_920_Diff__idemp,axiom,
! [A3: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B2 ) @ B2 )
= ( minus_minus_set_a @ A3 @ B2 ) ) ).
% Diff_idemp
thf(fact_921_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_922_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_923_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_924_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_925_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_926_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_927_inf__top__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X )
= X ) ).
% inf_top_left
thf(fact_928_inf__top__left,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ top_top_a_o @ X )
= X ) ).
% inf_top_left
thf(fact_929_inf__top__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ top_top_set_set_a @ X )
= X ) ).
% inf_top_left
thf(fact_930_inf__top__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ top_top_set_a )
= X ) ).
% inf_top_right
thf(fact_931_inf__top__right,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ X @ top_top_a_o )
= X ) ).
% inf_top_right
thf(fact_932_inf__top__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ top_top_set_set_a )
= X ) ).
% inf_top_right
thf(fact_933_inf__eq__top__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X @ Y )
= top_top_set_a )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_934_inf__eq__top__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( inf_inf_a_o @ X @ Y )
= top_top_a_o )
= ( ( X = top_top_a_o )
& ( Y = top_top_a_o ) ) ) ).
% inf_eq_top_iff
thf(fact_935_inf__eq__top__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( inf_inf_set_set_a @ X @ Y )
= top_top_set_set_a )
= ( ( X = top_top_set_set_a )
& ( Y = top_top_set_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_936_top__eq__inf__iff,axiom,
! [X: set_a,Y: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ X @ Y ) )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_937_top__eq__inf__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( top_top_a_o
= ( inf_inf_a_o @ X @ Y ) )
= ( ( X = top_top_a_o )
& ( Y = top_top_a_o ) ) ) ).
% top_eq_inf_iff
thf(fact_938_top__eq__inf__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( top_top_set_set_a
= ( inf_inf_set_set_a @ X @ Y ) )
= ( ( X = top_top_set_set_a )
& ( Y = top_top_set_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_939_inf__top_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= top_top_set_a )
= ( ( A = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_940_inf__top_Oeq__neutr__iff,axiom,
! [A: a > $o,B: a > $o] :
( ( ( inf_inf_a_o @ A @ B )
= top_top_a_o )
= ( ( A = top_top_a_o )
& ( B = top_top_a_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_941_inf__top_Oeq__neutr__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= top_top_set_set_a )
= ( ( A = top_top_set_set_a )
& ( B = top_top_set_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_942_inf__top_Oleft__neutral,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_943_inf__top_Oleft__neutral,axiom,
! [A: a > $o] :
( ( inf_inf_a_o @ top_top_a_o @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_944_inf__top_Oleft__neutral,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ top_top_set_set_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_945_inf__top_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A @ B ) )
= ( ( A = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_946_inf__top_Oneutr__eq__iff,axiom,
! [A: a > $o,B: a > $o] :
( ( top_top_a_o
= ( inf_inf_a_o @ A @ B ) )
= ( ( A = top_top_a_o )
& ( B = top_top_a_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_947_inf__top_Oneutr__eq__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( top_top_set_set_a
= ( inf_inf_set_set_a @ A @ B ) )
= ( ( A = top_top_set_set_a )
& ( B = top_top_set_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_948_inf__top_Oright__neutral,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ top_top_set_a )
= A ) ).
% inf_top.right_neutral
thf(fact_949_inf__top_Oright__neutral,axiom,
! [A: a > $o] :
( ( inf_inf_a_o @ A @ top_top_a_o )
= A ) ).
% inf_top.right_neutral
thf(fact_950_inf__top_Oright__neutral,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ top_top_set_set_a )
= A ) ).
% inf_top.right_neutral
thf(fact_951_Inf__top__conv_I2_J,axiom,
! [A3: set_set_a] :
( ( top_top_set_a
= ( comple6135023378680113637_set_a @ A3 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( X3 = top_top_set_a ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_952_Inf__top__conv_I2_J,axiom,
! [A3: set_a_o] :
( ( top_top_a_o
= ( complete_Inf_Inf_a_o @ A3 ) )
= ( ! [X3: a > $o] :
( ( member_a_o @ X3 @ A3 )
=> ( X3 = top_top_a_o ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_953_Inf__top__conv_I2_J,axiom,
! [A3: set_set_set_a] :
( ( top_top_set_set_a
= ( comple9105089376463352645_set_a @ A3 ) )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A3 )
=> ( X3 = top_top_set_set_a ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_954_Inf__top__conv_I1_J,axiom,
! [A3: set_set_a] :
( ( ( comple6135023378680113637_set_a @ A3 )
= top_top_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( X3 = top_top_set_a ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_955_Inf__top__conv_I1_J,axiom,
! [A3: set_a_o] :
( ( ( complete_Inf_Inf_a_o @ A3 )
= top_top_a_o )
= ( ! [X3: a > $o] :
( ( member_a_o @ X3 @ A3 )
=> ( X3 = top_top_a_o ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_956_Inf__top__conv_I1_J,axiom,
! [A3: set_set_set_a] :
( ( ( comple9105089376463352645_set_a @ A3 )
= top_top_set_set_a )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A3 )
=> ( X3 = top_top_set_set_a ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_957_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_958_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_959_Int__UNIV,axiom,
! [A3: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A3 @ B2 )
= top_top_set_a )
= ( ( A3 = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_960_Int__UNIV,axiom,
! [A3: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A3 @ B2 )
= top_top_set_set_a )
= ( ( A3 = top_top_set_set_a )
& ( B2 = top_top_set_set_a ) ) ) ).
% Int_UNIV
thf(fact_961_Diff__empty,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Diff_empty
thf(fact_962_empty__Diff,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A3 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_963_Diff__cancel,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ A3 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_964_Int__subset__iff,axiom,
! [C2: set_a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A3 @ B2 ) )
= ( ( ord_less_eq_set_a @ C2 @ A3 )
& ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_965_Int__insert__right__if1,axiom,
! [A: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A @ B2 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A3 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_966_Int__insert__right__if1,axiom,
! [A: a,A3: set_a,B2: set_a] :
( ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_967_Int__insert__right__if0,axiom,
! [A: set_a,A3: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A @ B2 ) )
= ( inf_inf_set_set_a @ A3 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_968_Int__insert__right__if0,axiom,
! [A: a,A3: set_a,B2: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A3 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_969_insert__inter__insert,axiom,
! [A: a,A3: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A3 ) @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A3 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_970_Int__insert__left__if1,axiom,
! [A: set_a,C2: set_set_a,B2: set_set_a] :
( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_971_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_972_Int__insert__left__if0,axiom,
! [A: set_a,C2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_set_a @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_973_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_974_insert__Diff1,axiom,
! [X: set_a,B2: set_set_a,A3: set_set_a] :
( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A3 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_975_insert__Diff1,axiom,
! [X: a,B2: set_a,A3: set_a] :
( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B2 )
= ( minus_minus_set_a @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_976_Diff__insert0,axiom,
! [X: set_a,A3: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A3 )
=> ( ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X @ B2 ) )
= ( minus_5736297505244876581_set_a @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_977_Diff__insert0,axiom,
! [X: a,A3: set_a,B2: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( minus_minus_set_a @ A3 @ ( insert_a @ X @ B2 ) )
= ( minus_minus_set_a @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_978_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_979_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_980_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_981_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_982_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_983_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_984_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_985_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_986_Un__Diff__cancel2,axiom,
! [B2: set_a,A3: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B2 @ A3 ) @ A3 )
= ( sup_sup_set_a @ B2 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_987_Un__Diff__cancel,axiom,
! [A3: set_a,B2: set_a] :
( ( sup_sup_set_a @ A3 @ ( minus_minus_set_a @ B2 @ A3 ) )
= ( sup_sup_set_a @ A3 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_988_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ ( uminus_uminus_set_a @ X ) )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_989_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_990_inf__compl__bot__right,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ ( uminus_uminus_set_a @ X ) ) )
= bot_bot_set_a ) ).
% inf_compl_bot_right
thf(fact_991_inf__compl__bot__left2,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left2
thf(fact_992_inf__compl__bot__left1,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ ( inf_inf_set_a @ X @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left1
thf(fact_993_Inf__UNIV,axiom,
( ( comple6135023378680113637_set_a @ top_top_set_set_a )
= bot_bot_set_a ) ).
% Inf_UNIV
thf(fact_994_Inf__empty,axiom,
( ( comple6135023378680113637_set_a @ bot_bot_set_set_a )
= top_top_set_a ) ).
% Inf_empty
thf(fact_995_Inf__empty,axiom,
( ( complete_Inf_Inf_a_o @ bot_bot_set_a_o )
= top_top_a_o ) ).
% Inf_empty
thf(fact_996_Inf__empty,axiom,
( ( comple9105089376463352645_set_a @ bot_bo3380559777022489994_set_a )
= top_top_set_set_a ) ).
% Inf_empty
thf(fact_997_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_a,Y: set_a] :
( ( uminus_uminus_set_a @ ( sup_sup_set_a @ X @ Y ) )
= ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_998_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_a,Y: set_a] :
( ( uminus_uminus_set_a @ ( inf_inf_set_a @ X @ Y ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_999_Diff__UNIV,axiom,
! [A3: set_set_a] :
( ( minus_5736297505244876581_set_a @ A3 @ top_top_set_set_a )
= bot_bot_set_set_a ) ).
% Diff_UNIV
thf(fact_1000_Diff__UNIV,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_1001_cInf__atLeastAtMost,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ Y @ X ) )
= Y ) ) ).
% cInf_atLeastAtMost
thf(fact_1002_Inf__atLeastAtMost,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ X @ Y ) )
= X ) ) ).
% Inf_atLeastAtMost
thf(fact_1003_disjoint__insert_I2_J,axiom,
! [A3: set_set_a,B: set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ B @ B2 ) ) )
= ( ~ ( member_set_a @ B @ A3 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1004_disjoint__insert_I2_J,axiom,
! [A3: set_a,B: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ ( insert_a @ B @ B2 ) ) )
= ( ~ ( member_a @ B @ A3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1005_disjoint__insert_I1_J,axiom,
! [B2: set_set_a,A: set_a,A3: set_set_a] :
( ( ( inf_inf_set_set_a @ B2 @ ( insert_set_a @ A @ A3 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B2 )
& ( ( inf_inf_set_set_a @ B2 @ A3 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1006_disjoint__insert_I1_J,axiom,
! [B2: set_a,A: a,A3: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A @ A3 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ B2 @ A3 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1007_insert__disjoint_I2_J,axiom,
! [A: set_a,A3: set_set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A3 ) @ B2 ) )
= ( ~ ( member_set_a @ A @ B2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1008_insert__disjoint_I2_J,axiom,
! [A: a,A3: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A3 ) @ B2 ) )
= ( ~ ( member_a @ A @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1009_insert__disjoint_I1_J,axiom,
! [A: set_a,A3: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A3 ) @ B2 )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B2 )
& ( ( inf_inf_set_set_a @ A3 @ B2 )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1010_insert__disjoint_I1_J,axiom,
! [A: a,A3: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A3 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ A3 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1011_Diff__eq__empty__iff,axiom,
! [A3: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A3 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A3 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1012_insert__Diff__single,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_1013_ivl__diff,axiom,
! [I: a,N: a,M2: a] :
( ( ord_less_eq_a @ I @ N )
=> ( ( minus_minus_set_a @ ( set_or5139330845457685135Than_a @ I @ M2 ) @ ( set_or5139330845457685135Than_a @ I @ N ) )
= ( set_or5139330845457685135Than_a @ N @ M2 ) ) ) ).
% ivl_diff
thf(fact_1014_cInf__atLeastLessThan,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_set_a @ Y @ X )
=> ( ( comple6135023378680113637_set_a @ ( set_or2348907005316661231_set_a @ Y @ X ) )
= Y ) ) ).
% cInf_atLeastLessThan
thf(fact_1015_Inf__atLeastLessThan,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( comple6135023378680113637_set_a @ ( set_or2348907005316661231_set_a @ X @ Y ) )
= X ) ) ).
% Inf_atLeastLessThan
thf(fact_1016_Diff__disjoint,axiom,
! [A3: set_a,B2: set_a] :
( ( inf_inf_set_a @ A3 @ ( minus_minus_set_a @ B2 @ A3 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_1017_Inf__atMost,axiom,
! [X: set_a] :
( ( comple6135023378680113637_set_a @ ( set_ord_atMost_set_a @ X ) )
= bot_bot_set_a ) ).
% Inf_atMost
thf(fact_1018_Compl__disjoint2,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ A3 ) @ A3 )
= bot_bot_set_a ) ).
% Compl_disjoint2
thf(fact_1019_Compl__disjoint,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ ( uminus_uminus_set_a @ A3 ) )
= bot_bot_set_a ) ).
% Compl_disjoint
thf(fact_1020_lessThan__minus__lessThan,axiom,
! [N: a,M2: a] :
( ( minus_minus_set_a @ ( set_ord_lessThan_a @ N ) @ ( set_ord_lessThan_a @ M2 ) )
= ( set_or5139330845457685135Than_a @ M2 @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_1021_Diff__Compl,axiom,
! [A3: set_a,B2: set_a] :
( ( minus_minus_set_a @ A3 @ ( uminus_uminus_set_a @ B2 ) )
= ( inf_inf_set_a @ A3 @ B2 ) ) ).
% Diff_Compl
thf(fact_1022_Compl__Diff__eq,axiom,
! [A3: set_a,B2: set_a] :
( ( uminus_uminus_set_a @ ( minus_minus_set_a @ A3 @ B2 ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A3 ) @ B2 ) ) ).
% Compl_Diff_eq
thf(fact_1023_single__Diff__lessThan,axiom,
! [K: a] :
( ( minus_minus_set_a @ ( insert_a @ K @ bot_bot_set_a ) @ ( set_ord_lessThan_a @ K ) )
= ( insert_a @ K @ bot_bot_set_a ) ) ).
% single_Diff_lessThan
thf(fact_1024_Inter__Un__subset,axiom,
! [A3: set_set_a,B2: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A3 ) @ ( comple6135023378680113637_set_a @ B2 ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A3 @ B2 ) ) ) ).
% Inter_Un_subset
thf(fact_1025_less__eq__Inf__inter,axiom,
! [A3: set_set_a,B2: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A3 ) @ ( comple6135023378680113637_set_a @ B2 ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A3 @ B2 ) ) ) ).
% less_eq_Inf_inter
thf(fact_1026_mono__Int,axiom,
! [F: set_a > set_a,A3: set_a,B2: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A3 @ B2 ) ) @ ( inf_inf_set_a @ ( F @ A3 ) @ ( F @ B2 ) ) ) ) ).
% mono_Int
thf(fact_1027_monotoneI,axiom,
! [Orda: a > a > $o,Ordb: a > a > $o,F: a > a] :
( ! [X2: a,Y3: a] :
( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_a_a @ top_top_set_a @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_1028_monotoneI,axiom,
! [Orda: set_a > set_a > $o,Ordb: set_a > set_a > $o,F: set_a > set_a] :
( ! [X2: set_a,Y3: set_a] :
( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto7172710143293369831_set_a @ top_top_set_set_a @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_1029_monotoneD,axiom,
! [Orda: a > a > $o,Ordb: a > a > $o,F: a > a,X: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_1030_monotoneD,axiom,
! [Orda: set_a > set_a > $o,Ordb: set_a > set_a > $o,F: set_a > set_a,X: set_a,Y: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_1031_mono__inf,axiom,
! [F: set_a > set_a,A3: set_a,B2: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A3 @ B2 ) ) @ ( inf_inf_set_a @ ( F @ A3 ) @ ( F @ B2 ) ) ) ) ).
% mono_inf
thf(fact_1032_monotone__on__subset,axiom,
! [A3: set_set_a,Orda: set_a > set_a > $o,Ordb: set_a > set_a > $o,F: set_a > set_a,B2: set_set_a] :
( ( monoto7172710143293369831_set_a @ A3 @ Orda @ Ordb @ F )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A3 )
=> ( monoto7172710143293369831_set_a @ B2 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_1033_monotone__on__subset,axiom,
! [A3: set_a,Orda: a > a > $o,Ordb: a > a > $o,F: a > a,B2: set_a] :
( ( monotone_on_a_a @ A3 @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( monotone_on_a_a @ B2 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_1034_Inf__greatest,axiom,
! [A3: set_set_a,Z: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ Z @ X2 ) )
=> ( ord_less_eq_set_a @ Z @ ( comple6135023378680113637_set_a @ A3 ) ) ) ).
% Inf_greatest
thf(fact_1035_le__Inf__iff,axiom,
! [B: set_a,A3: set_set_a] :
( ( ord_less_eq_set_a @ B @ ( comple6135023378680113637_set_a @ A3 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( ord_less_eq_set_a @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_1036_Inf__lower2,axiom,
! [U: set_a,A3: set_set_a,V: set_a] :
( ( member_set_a @ U @ A3 )
=> ( ( ord_less_eq_set_a @ U @ V )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A3 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1037_Inf__lower,axiom,
! [X: set_a,A3: set_set_a] :
( ( member_set_a @ X @ A3 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A3 ) @ X ) ) ).
% Inf_lower
thf(fact_1038_Inf__mono,axiom,
! [B2: set_set_a,A3: set_set_a] :
( ! [B5: set_a] :
( ( member_set_a @ B5 @ B2 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
& ( ord_less_eq_set_a @ X4 @ B5 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A3 ) @ ( comple6135023378680113637_set_a @ B2 ) ) ) ).
% Inf_mono
thf(fact_1039_Inf__eqI,axiom,
! [A3: set_set_a,X: set_a] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A3 )
=> ( ord_less_eq_set_a @ X @ I2 ) )
=> ( ! [Y3: set_a] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A3 )
=> ( ord_less_eq_set_a @ Y3 @ I3 ) )
=> ( ord_less_eq_set_a @ Y3 @ X ) )
=> ( ( comple6135023378680113637_set_a @ A3 )
= X ) ) ) ).
% Inf_eqI
thf(fact_1040_mono__onI,axiom,
! [A3: set_a,F: a > set_a] :
( ! [R4: a,S4: a] :
( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S4 @ A3 )
=> ( ( ord_less_eq_a @ R4 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_set_a @ A3 @ ord_less_eq_a @ ord_less_eq_set_a @ F ) ) ).
% mono_onI
thf(fact_1041_mono__onI,axiom,
! [A3: set_a,F: a > $o > a] :
( ! [R4: a,S4: a] :
( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S4 @ A3 )
=> ( ( ord_less_eq_a @ R4 @ S4 )
=> ( ord_less_eq_o_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_o_a @ A3 @ ord_less_eq_a @ ord_less_eq_o_a @ F ) ) ).
% mono_onI
thf(fact_1042_mono__onI,axiom,
! [A3: set_set_a,F: set_a > a] :
( ! [R4: set_a,S4: set_a] :
( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S4 @ A3 )
=> ( ( ord_less_eq_set_a @ R4 @ S4 )
=> ( ord_less_eq_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_set_a_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_a @ F ) ) ).
% mono_onI
thf(fact_1043_mono__onI,axiom,
! [A3: set_set_a,F: set_a > set_a] :
( ! [R4: set_a,S4: set_a] :
( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S4 @ A3 )
=> ( ( ord_less_eq_set_a @ R4 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monoto7172710143293369831_set_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% mono_onI
thf(fact_1044_mono__onI,axiom,
! [A3: set_set_a,F: set_a > $o > a] :
( ! [R4: set_a,S4: set_a] :
( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S4 @ A3 )
=> ( ( ord_less_eq_set_a @ R4 @ S4 )
=> ( ord_less_eq_o_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monoto9194799565806279554_a_o_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_o_a @ F ) ) ).
% mono_onI
thf(fact_1045_mono__onI,axiom,
! [A3: set_o_a,F: ( $o > a ) > a] :
( ! [R4: $o > a,S4: $o > a] :
( ( member_o_a @ R4 @ A3 )
=> ( ( member_o_a @ S4 @ A3 )
=> ( ( ord_less_eq_o_a @ R4 @ S4 )
=> ( ord_less_eq_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_o_a_a @ A3 @ ord_less_eq_o_a @ ord_less_eq_a @ F ) ) ).
% mono_onI
thf(fact_1046_mono__onI,axiom,
! [A3: set_o_a,F: ( $o > a ) > set_a] :
( ! [R4: $o > a,S4: $o > a] :
( ( member_o_a @ R4 @ A3 )
=> ( ( member_o_a @ S4 @ A3 )
=> ( ( ord_less_eq_o_a @ R4 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monoto888385181794500650_set_a @ A3 @ ord_less_eq_o_a @ ord_less_eq_set_a @ F ) ) ).
% mono_onI
thf(fact_1047_mono__onI,axiom,
! [A3: set_o_a,F: ( $o > a ) > $o > a] :
( ! [R4: $o > a,S4: $o > a] :
( ( member_o_a @ R4 @ A3 )
=> ( ( member_o_a @ S4 @ A3 )
=> ( ( ord_less_eq_o_a @ R4 @ S4 )
=> ( ord_less_eq_o_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_o_a_o_a @ A3 @ ord_less_eq_o_a @ ord_less_eq_o_a @ F ) ) ).
% mono_onI
thf(fact_1048_mono__onI,axiom,
! [A3: set_a,F: a > a] :
( ! [R4: a,S4: a] :
( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S4 @ A3 )
=> ( ( ord_less_eq_a @ R4 @ S4 )
=> ( ord_less_eq_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_a @ A3 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% mono_onI
thf(fact_1049_mono__onD,axiom,
! [A3: set_a,F: a > set_a,R3: a,S3: a] :
( ( monotone_on_a_set_a @ A3 @ ord_less_eq_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( ord_less_eq_a @ R3 @ S3 )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1050_mono__onD,axiom,
! [A3: set_a,F: a > $o > a,R3: a,S3: a] :
( ( monotone_on_a_o_a @ A3 @ ord_less_eq_a @ ord_less_eq_o_a @ F )
=> ( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( ord_less_eq_a @ R3 @ S3 )
=> ( ord_less_eq_o_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1051_mono__onD,axiom,
! [A3: set_set_a,F: set_a > a,R3: set_a,S3: set_a] :
( ( monotone_on_set_a_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_a @ F )
=> ( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( ord_less_eq_set_a @ R3 @ S3 )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1052_mono__onD,axiom,
! [A3: set_set_a,F: set_a > set_a,R3: set_a,S3: set_a] :
( ( monoto7172710143293369831_set_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( ord_less_eq_set_a @ R3 @ S3 )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1053_mono__onD,axiom,
! [A3: set_set_a,F: set_a > $o > a,R3: set_a,S3: set_a] :
( ( monoto9194799565806279554_a_o_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_o_a @ F )
=> ( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( ord_less_eq_set_a @ R3 @ S3 )
=> ( ord_less_eq_o_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1054_mono__onD,axiom,
! [A3: set_o_a,F: ( $o > a ) > a,R3: $o > a,S3: $o > a] :
( ( monotone_on_o_a_a @ A3 @ ord_less_eq_o_a @ ord_less_eq_a @ F )
=> ( ( member_o_a @ R3 @ A3 )
=> ( ( member_o_a @ S3 @ A3 )
=> ( ( ord_less_eq_o_a @ R3 @ S3 )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1055_mono__onD,axiom,
! [A3: set_o_a,F: ( $o > a ) > set_a,R3: $o > a,S3: $o > a] :
( ( monoto888385181794500650_set_a @ A3 @ ord_less_eq_o_a @ ord_less_eq_set_a @ F )
=> ( ( member_o_a @ R3 @ A3 )
=> ( ( member_o_a @ S3 @ A3 )
=> ( ( ord_less_eq_o_a @ R3 @ S3 )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1056_mono__onD,axiom,
! [A3: set_o_a,F: ( $o > a ) > $o > a,R3: $o > a,S3: $o > a] :
( ( monotone_on_o_a_o_a @ A3 @ ord_less_eq_o_a @ ord_less_eq_o_a @ F )
=> ( ( member_o_a @ R3 @ A3 )
=> ( ( member_o_a @ S3 @ A3 )
=> ( ( ord_less_eq_o_a @ R3 @ S3 )
=> ( ord_less_eq_o_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1057_mono__onD,axiom,
! [A3: set_a,F: a > a,R3: a,S3: a] :
( ( monotone_on_a_a @ A3 @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( ord_less_eq_a @ R3 @ S3 )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1058_ord_Omono__on__def,axiom,
! [A3: set_set_a,Less_eq2: set_a > set_a > $o,F: set_a > a] :
( ( monotone_on_set_a_a @ A3 @ Less_eq2 @ ord_less_eq_a @ F )
= ( ! [R5: set_a,S5: set_a] :
( ( ( member_set_a @ R5 @ A3 )
& ( member_set_a @ S5 @ A3 )
& ( Less_eq2 @ R5 @ S5 ) )
=> ( ord_less_eq_a @ ( F @ R5 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1059_ord_Omono__on__def,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > set_a] :
( ( monotone_on_a_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F )
= ( ! [R5: a,S5: a] :
( ( ( member_a @ R5 @ A3 )
& ( member_a @ S5 @ A3 )
& ( Less_eq2 @ R5 @ S5 ) )
=> ( ord_less_eq_set_a @ ( F @ R5 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1060_ord_Omono__on__def,axiom,
! [A3: set_set_a,Less_eq2: set_a > set_a > $o,F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F )
= ( ! [R5: set_a,S5: set_a] :
( ( ( member_set_a @ R5 @ A3 )
& ( member_set_a @ S5 @ A3 )
& ( Less_eq2 @ R5 @ S5 ) )
=> ( ord_less_eq_set_a @ ( F @ R5 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1061_ord_Omono__on__def,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > $o > a] :
( ( monotone_on_a_o_a @ A3 @ Less_eq2 @ ord_less_eq_o_a @ F )
= ( ! [R5: a,S5: a] :
( ( ( member_a @ R5 @ A3 )
& ( member_a @ S5 @ A3 )
& ( Less_eq2 @ R5 @ S5 ) )
=> ( ord_less_eq_o_a @ ( F @ R5 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1062_ord_Omono__on__def,axiom,
! [A3: set_set_a,Less_eq2: set_a > set_a > $o,F: set_a > $o > a] :
( ( monoto9194799565806279554_a_o_a @ A3 @ Less_eq2 @ ord_less_eq_o_a @ F )
= ( ! [R5: set_a,S5: set_a] :
( ( ( member_set_a @ R5 @ A3 )
& ( member_set_a @ S5 @ A3 )
& ( Less_eq2 @ R5 @ S5 ) )
=> ( ord_less_eq_o_a @ ( F @ R5 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1063_ord_Omono__on__def,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > a] :
( ( monotone_on_a_a @ A3 @ Less_eq2 @ ord_less_eq_a @ F )
= ( ! [R5: a,S5: a] :
( ( ( member_a @ R5 @ A3 )
& ( member_a @ S5 @ A3 )
& ( Less_eq2 @ R5 @ S5 ) )
=> ( ord_less_eq_a @ ( F @ R5 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1064_ord_Omono__onI,axiom,
! [A3: set_set_a,Less_eq2: set_a > set_a > $o,F: set_a > a] :
( ! [R4: set_a,S4: set_a] :
( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S4 @ A3 )
=> ( ( Less_eq2 @ R4 @ S4 )
=> ( ord_less_eq_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_set_a_a @ A3 @ Less_eq2 @ ord_less_eq_a @ F ) ) ).
% ord.mono_onI
thf(fact_1065_ord_Omono__onI,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > set_a] :
( ! [R4: a,S4: a] :
( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S4 @ A3 )
=> ( ( Less_eq2 @ R4 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_1066_ord_Omono__onI,axiom,
! [A3: set_set_a,Less_eq2: set_a > set_a > $o,F: set_a > set_a] :
( ! [R4: set_a,S4: set_a] :
( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S4 @ A3 )
=> ( ( Less_eq2 @ R4 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monoto7172710143293369831_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_1067_ord_Omono__onI,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > $o > a] :
( ! [R4: a,S4: a] :
( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S4 @ A3 )
=> ( ( Less_eq2 @ R4 @ S4 )
=> ( ord_less_eq_o_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_o_a @ A3 @ Less_eq2 @ ord_less_eq_o_a @ F ) ) ).
% ord.mono_onI
thf(fact_1068_ord_Omono__onI,axiom,
! [A3: set_set_a,Less_eq2: set_a > set_a > $o,F: set_a > $o > a] :
( ! [R4: set_a,S4: set_a] :
( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S4 @ A3 )
=> ( ( Less_eq2 @ R4 @ S4 )
=> ( ord_less_eq_o_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monoto9194799565806279554_a_o_a @ A3 @ Less_eq2 @ ord_less_eq_o_a @ F ) ) ).
% ord.mono_onI
thf(fact_1069_ord_Omono__onI,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > a] :
( ! [R4: a,S4: a] :
( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S4 @ A3 )
=> ( ( Less_eq2 @ R4 @ S4 )
=> ( ord_less_eq_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_a @ A3 @ Less_eq2 @ ord_less_eq_a @ F ) ) ).
% ord.mono_onI
thf(fact_1070_ord_Omono__onD,axiom,
! [A3: set_set_a,Less_eq2: set_a > set_a > $o,F: set_a > a,R3: set_a,S3: set_a] :
( ( monotone_on_set_a_a @ A3 @ Less_eq2 @ ord_less_eq_a @ F )
=> ( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R3 @ S3 )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1071_ord_Omono__onD,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > set_a,R3: a,S3: a] :
( ( monotone_on_a_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F )
=> ( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R3 @ S3 )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1072_ord_Omono__onD,axiom,
! [A3: set_set_a,Less_eq2: set_a > set_a > $o,F: set_a > set_a,R3: set_a,S3: set_a] :
( ( monoto7172710143293369831_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F )
=> ( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R3 @ S3 )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1073_ord_Omono__onD,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > $o > a,R3: a,S3: a] :
( ( monotone_on_a_o_a @ A3 @ Less_eq2 @ ord_less_eq_o_a @ F )
=> ( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R3 @ S3 )
=> ( ord_less_eq_o_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1074_ord_Omono__onD,axiom,
! [A3: set_set_a,Less_eq2: set_a > set_a > $o,F: set_a > $o > a,R3: set_a,S3: set_a] :
( ( monoto9194799565806279554_a_o_a @ A3 @ Less_eq2 @ ord_less_eq_o_a @ F )
=> ( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R3 @ S3 )
=> ( ord_less_eq_o_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1075_ord_Omono__onD,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > a,R3: a,S3: a] :
( ( monotone_on_a_a @ A3 @ Less_eq2 @ ord_less_eq_a @ F )
=> ( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R3 @ S3 )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1076_Inter__lower,axiom,
! [B2: set_a,A3: set_set_a] :
( ( member_set_a @ B2 @ A3 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A3 ) @ B2 ) ) ).
% Inter_lower
thf(fact_1077_Inter__greatest,axiom,
! [A3: set_set_a,C2: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A3 )
=> ( ord_less_eq_set_a @ C2 @ X7 ) )
=> ( ord_less_eq_set_a @ C2 @ ( comple6135023378680113637_set_a @ A3 ) ) ) ).
% Inter_greatest
thf(fact_1078_Inter__anti__mono,axiom,
! [B2: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A3 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A3 ) @ ( comple6135023378680113637_set_a @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_1079_ord_Ostrict__mono__onD,axiom,
! [A3: set_set_a,Less: set_a > set_a > $o,F: set_a > a,R3: set_a,S3: set_a] :
( ( monotone_on_set_a_a @ A3 @ Less @ ord_less_a @ F )
=> ( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( Less @ R3 @ S3 )
=> ( ord_less_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1080_ord_Ostrict__mono__onD,axiom,
! [A3: set_a,Less: a > a > $o,F: a > a,R3: a,S3: a] :
( ( monotone_on_a_a @ A3 @ Less @ ord_less_a @ F )
=> ( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( Less @ R3 @ S3 )
=> ( ord_less_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1081_ord_Ostrict__mono__onD,axiom,
! [A3: set_a,Less: a > a > $o,F: a > set_a,R3: a,S3: a] :
( ( monotone_on_a_set_a @ A3 @ Less @ ord_less_set_a @ F )
=> ( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( Less @ R3 @ S3 )
=> ( ord_less_set_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1082_ord_Ostrict__mono__onD,axiom,
! [A3: set_set_a,Less: set_a > set_a > $o,F: set_a > set_a,R3: set_a,S3: set_a] :
( ( monoto7172710143293369831_set_a @ A3 @ Less @ ord_less_set_a @ F )
=> ( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( Less @ R3 @ S3 )
=> ( ord_less_set_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1083_ord_Ostrict__mono__onI,axiom,
! [A3: set_set_a,Less: set_a > set_a > $o,F: set_a > a] :
( ! [R4: set_a,S4: set_a] :
( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S4 @ A3 )
=> ( ( Less @ R4 @ S4 )
=> ( ord_less_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_set_a_a @ A3 @ Less @ ord_less_a @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1084_ord_Ostrict__mono__onI,axiom,
! [A3: set_a,Less: a > a > $o,F: a > a] :
( ! [R4: a,S4: a] :
( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S4 @ A3 )
=> ( ( Less @ R4 @ S4 )
=> ( ord_less_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_a @ A3 @ Less @ ord_less_a @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1085_ord_Ostrict__mono__onI,axiom,
! [A3: set_a,Less: a > a > $o,F: a > set_a] :
( ! [R4: a,S4: a] :
( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S4 @ A3 )
=> ( ( Less @ R4 @ S4 )
=> ( ord_less_set_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_set_a @ A3 @ Less @ ord_less_set_a @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1086_ord_Ostrict__mono__onI,axiom,
! [A3: set_set_a,Less: set_a > set_a > $o,F: set_a > set_a] :
( ! [R4: set_a,S4: set_a] :
( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S4 @ A3 )
=> ( ( Less @ R4 @ S4 )
=> ( ord_less_set_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monoto7172710143293369831_set_a @ A3 @ Less @ ord_less_set_a @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1087_ord_Ostrict__mono__on__def,axiom,
! [A3: set_set_a,Less: set_a > set_a > $o,F: set_a > a] :
( ( monotone_on_set_a_a @ A3 @ Less @ ord_less_a @ F )
= ( ! [R5: set_a,S5: set_a] :
( ( ( member_set_a @ R5 @ A3 )
& ( member_set_a @ S5 @ A3 )
& ( Less @ R5 @ S5 ) )
=> ( ord_less_a @ ( F @ R5 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_1088_ord_Ostrict__mono__on__def,axiom,
! [A3: set_a,Less: a > a > $o,F: a > a] :
( ( monotone_on_a_a @ A3 @ Less @ ord_less_a @ F )
= ( ! [R5: a,S5: a] :
( ( ( member_a @ R5 @ A3 )
& ( member_a @ S5 @ A3 )
& ( Less @ R5 @ S5 ) )
=> ( ord_less_a @ ( F @ R5 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_1089_ord_Ostrict__mono__on__def,axiom,
! [A3: set_a,Less: a > a > $o,F: a > set_a] :
( ( monotone_on_a_set_a @ A3 @ Less @ ord_less_set_a @ F )
= ( ! [R5: a,S5: a] :
( ( ( member_a @ R5 @ A3 )
& ( member_a @ S5 @ A3 )
& ( Less @ R5 @ S5 ) )
=> ( ord_less_set_a @ ( F @ R5 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_1090_ord_Ostrict__mono__on__def,axiom,
! [A3: set_set_a,Less: set_a > set_a > $o,F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ A3 @ Less @ ord_less_set_a @ F )
= ( ! [R5: set_a,S5: set_a] :
( ( ( member_set_a @ R5 @ A3 )
& ( member_set_a @ S5 @ A3 )
& ( Less @ R5 @ S5 ) )
=> ( ord_less_set_a @ ( F @ R5 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_1091_strict__mono__onD,axiom,
! [A3: set_a,F: a > a,R3: a,S3: a] :
( ( monotone_on_a_a @ A3 @ ord_less_a @ ord_less_a @ F )
=> ( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( ord_less_a @ R3 @ S3 )
=> ( ord_less_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_1092_strict__mono__onD,axiom,
! [A3: set_a,F: a > set_a,R3: a,S3: a] :
( ( monotone_on_a_set_a @ A3 @ ord_less_a @ ord_less_set_a @ F )
=> ( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( ord_less_a @ R3 @ S3 )
=> ( ord_less_set_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_1093_strict__mono__onD,axiom,
! [A3: set_set_a,F: set_a > a,R3: set_a,S3: set_a] :
( ( monotone_on_set_a_a @ A3 @ ord_less_set_a @ ord_less_a @ F )
=> ( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( ord_less_set_a @ R3 @ S3 )
=> ( ord_less_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_1094_strict__mono__onD,axiom,
! [A3: set_set_a,F: set_a > set_a,R3: set_a,S3: set_a] :
( ( monoto7172710143293369831_set_a @ A3 @ ord_less_set_a @ ord_less_set_a @ F )
=> ( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( ord_less_set_a @ R3 @ S3 )
=> ( ord_less_set_a @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_1095_strict__mono__onI,axiom,
! [A3: set_a,F: a > a] :
( ! [R4: a,S4: a] :
( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S4 @ A3 )
=> ( ( ord_less_a @ R4 @ S4 )
=> ( ord_less_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_a @ A3 @ ord_less_a @ ord_less_a @ F ) ) ).
% strict_mono_onI
thf(fact_1096_strict__mono__onI,axiom,
! [A3: set_a,F: a > set_a] :
( ! [R4: a,S4: a] :
( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S4 @ A3 )
=> ( ( ord_less_a @ R4 @ S4 )
=> ( ord_less_set_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_set_a @ A3 @ ord_less_a @ ord_less_set_a @ F ) ) ).
% strict_mono_onI
thf(fact_1097_strict__mono__onI,axiom,
! [A3: set_set_a,F: set_a > a] :
( ! [R4: set_a,S4: set_a] :
( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S4 @ A3 )
=> ( ( ord_less_set_a @ R4 @ S4 )
=> ( ord_less_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_set_a_a @ A3 @ ord_less_set_a @ ord_less_a @ F ) ) ).
% strict_mono_onI
thf(fact_1098_strict__mono__onI,axiom,
! [A3: set_set_a,F: set_a > set_a] :
( ! [R4: set_a,S4: set_a] :
( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S4 @ A3 )
=> ( ( ord_less_set_a @ R4 @ S4 )
=> ( ord_less_set_a @ ( F @ R4 ) @ ( F @ S4 ) ) ) ) )
=> ( monoto7172710143293369831_set_a @ A3 @ ord_less_set_a @ ord_less_set_a @ F ) ) ).
% strict_mono_onI
thf(fact_1099_strict__mono__on__eqD,axiom,
! [A3: set_a,F: a > a,X: a,Y: a] :
( ( monotone_on_a_a @ A3 @ ord_less_a @ ord_less_a @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_a @ X @ A3 )
=> ( ( member_a @ Y @ A3 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_1100_strict__mono__on__eqD,axiom,
! [A3: set_a,F: a > set_a,X: a,Y: a] :
( ( monotone_on_a_set_a @ A3 @ ord_less_a @ ord_less_set_a @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_a @ X @ A3 )
=> ( ( member_a @ Y @ A3 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_1101_psubset__imp__ex__mem,axiom,
! [A3: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A3 @ B2 )
=> ? [B5: set_a] : ( member_set_a @ B5 @ ( minus_5736297505244876581_set_a @ B2 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1102_psubset__imp__ex__mem,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_set_a @ A3 @ B2 )
=> ? [B5: a] : ( member_a @ B5 @ ( minus_minus_set_a @ B2 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1103_diff__eq,axiom,
( minus_minus_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( inf_inf_set_a @ X3 @ ( uminus_uminus_set_a @ Y4 ) ) ) ) ).
% diff_eq
thf(fact_1104_Diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A2: set_a,B3: set_a] : ( inf_inf_set_a @ A2 @ ( uminus_uminus_set_a @ B3 ) ) ) ) ).
% Diff_eq
thf(fact_1105_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_1106_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_1107_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_1108_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_1109_inf_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_1110_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_1111_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_1112_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_1113_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_a,K: set_a,A: set_a,B: set_a] :
( ( A3
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A3 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1114_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_a,K: set_a,B: set_a,A: set_a] :
( ( B2
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1115_inf_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_1116_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_1117_IntE,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A3 )
=> ~ ( member_set_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_1118_IntE,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B2 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ~ ( member_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_1119_DiffE,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1120_DiffE,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B2 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1121_IntD1,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B2 ) )
=> ( member_set_a @ C @ A3 ) ) ).
% IntD1
thf(fact_1122_IntD1,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B2 ) )
=> ( member_a @ C @ A3 ) ) ).
% IntD1
thf(fact_1123_IntD2,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B2 ) )
=> ( member_set_a @ C @ B2 ) ) ).
% IntD2
thf(fact_1124_IntD2,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B2 ) )
=> ( member_a @ C @ B2 ) ) ).
% IntD2
thf(fact_1125_DiffD1,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B2 ) )
=> ( member_set_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_1126_DiffD1,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B2 ) )
=> ( member_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_1127_DiffD2,axiom,
! [C: set_a,A3: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B2 ) )
=> ~ ( member_set_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_1128_DiffD2,axiom,
! [C: a,A3: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_1129_Int__Diff,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A3 @ ( minus_minus_set_a @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_1130_Diff__Int2,axiom,
! [A3: set_a,C2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A3 @ C2 ) @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A3 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1131_Int__assoc,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_1132_Int__absorb,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_1133_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A2 ) ) ) ).
% Int_commute
thf(fact_1134_Diff__Diff__Int,axiom,
! [A3: set_a,B2: set_a] :
( ( minus_minus_set_a @ A3 @ ( minus_minus_set_a @ A3 @ B2 ) )
= ( inf_inf_set_a @ A3 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1135_Int__left__absorb,axiom,
! [A3: set_a,B2: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B2 ) )
= ( inf_inf_set_a @ A3 @ B2 ) ) ).
% Int_left_absorb
thf(fact_1136_Diff__Int__distrib,axiom,
! [C2: set_a,A3: set_a,B2: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A3 @ B2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A3 ) @ ( inf_inf_set_a @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1137_Int__left__commute,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_1138_Diff__Int__distrib2,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A3 @ B2 ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A3 @ C2 ) @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1139_Un__Diff,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ C2 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ C2 ) @ ( minus_minus_set_a @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_1140_Un__Int__distrib2,axiom,
! [B2: set_a,C2: set_a,A3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B2 @ C2 ) @ A3 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B2 @ A3 ) @ ( sup_sup_set_a @ C2 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_1141_Int__Un__distrib2,axiom,
! [B2: set_a,C2: set_a,A3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A3 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B2 @ A3 ) @ ( inf_inf_set_a @ C2 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_1142_Un__Int__distrib,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ ( sup_sup_set_a @ A3 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1143_Int__Un__distrib,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ A3 @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ ( inf_inf_set_a @ A3 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1144_Un__Int__crazy,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ ( inf_inf_set_a @ B2 @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A3 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A3 @ B2 ) @ ( sup_sup_set_a @ B2 @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_1145_double__diff,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1146_Diff__subset,axiom,
! [A3: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B2 ) @ A3 ) ).
% Diff_subset
thf(fact_1147_Diff__mono,axiom,
! [A3: set_a,C2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B2 ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1148_Int__Collect__mono,axiom,
! [A3: set_set_a,B2: set_set_a,P2: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ ( collect_set_a @ P2 ) ) @ ( inf_inf_set_set_a @ B2 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1149_Int__Collect__mono,axiom,
! [A3: set_a,B2: set_a,P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P2 ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1150_Int__greatest,axiom,
! [C2: set_a,A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1151_Int__absorb2,axiom,
! [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
=> ( ( inf_inf_set_a @ A3 @ B2 )
= A3 ) ) ).
% Int_absorb2
thf(fact_1152_Int__absorb1,axiom,
! [B2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1153_Int__lower2,axiom,
! [A3: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1154_Int__lower1,axiom,
! [A3: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ A3 ) ).
% Int_lower1
thf(fact_1155_Int__mono,axiom,
! [A3: set_a,C2: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1156_Int__UNIV__right,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ top_top_set_a )
= A3 ) ).
% Int_UNIV_right
thf(fact_1157_Int__UNIV__right,axiom,
! [A3: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ top_top_set_set_a )
= A3 ) ).
% Int_UNIV_right
thf(fact_1158_Int__UNIV__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1159_Int__UNIV__left,axiom,
! [B2: set_set_a] :
( ( inf_inf_set_set_a @ top_top_set_set_a @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1160_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1161_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1162_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1163_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1164_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1165_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1166_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1167_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1168_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1169_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1170_boolean__algebra_Oconj__one__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ top_top_set_a )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_1171_boolean__algebra_Oconj__one__right,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ X @ top_top_a_o )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_1172_boolean__algebra_Oconj__one__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ top_top_set_set_a )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_1173_inf_Ostrict__coboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1174_inf_Ostrict__coboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_set_a @ A @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1175_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_1176_inf_Ostrict__boundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1177_inf_Oabsorb4,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_1178_inf_Oabsorb3,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_1179_less__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_set_a @ B @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_1180_less__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_set_a @ A @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_1181_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1182_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1183_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_1184_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_1185_inf_Ocobounded2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1186_inf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1187_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_1188_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1189_inf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1190_inf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1191_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1192_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1193_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1194_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1195_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1196_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1197_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_1198_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_1199_le__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_1200_le__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_1201_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1202_le__infI,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_1203_le__infE,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B ) ) ) ).
% le_infE
thf(fact_1204_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1205_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1206_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1207_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1208_cInf__eq__minimum,axiom,
! [Z: set_a,X6: set_set_a] :
( ( member_set_a @ Z @ X6 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ Z @ X2 ) )
=> ( ( comple6135023378680113637_set_a @ X6 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1209_Diff__Un,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ A3 @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A3 @ B2 ) @ ( minus_minus_set_a @ A3 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1210_Diff__Int,axiom,
! [A3: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ A3 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ B2 ) @ ( minus_minus_set_a @ A3 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1211_Int__Diff__Un,axiom,
! [A3: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ ( minus_minus_set_a @ A3 @ B2 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1212_Un__Diff__Int,axiom,
! [A3: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ B2 ) @ ( inf_inf_set_a @ A3 @ B2 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1213_Int__insert__right,axiom,
! [A: set_a,A3: set_set_a,B2: set_set_a] :
( ( ( member_set_a @ A @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A @ B2 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A3 @ B2 ) ) ) )
& ( ~ ( member_set_a @ A @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A @ B2 ) )
= ( inf_inf_set_set_a @ A3 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1214_Int__insert__right,axiom,
! [A: a,A3: set_a,B2: set_a] :
( ( ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A3 @ B2 ) ) ) )
& ( ~ ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1215_Int__insert__left,axiom,
! [A: set_a,C2: set_set_a,B2: set_set_a] :
( ( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C2 ) ) ) )
& ( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1216_Int__insert__left,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1217_insert__Diff__if,axiom,
! [X: set_a,B2: set_set_a,A3: set_set_a] :
( ( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A3 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A3 @ B2 ) ) )
& ( ~ ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A3 ) @ B2 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A3 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1218_insert__Diff__if,axiom,
! [X: a,B2: set_a,A3: set_a] :
( ( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B2 )
= ( minus_minus_set_a @ A3 @ B2 ) ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B2 )
= ( insert_a @ X @ ( minus_minus_set_a @ A3 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1219_Inter__empty,axiom,
( ( comple6135023378680113637_set_a @ bot_bot_set_set_a )
= top_top_set_a ) ).
% Inter_empty
thf(fact_1220_Inter__empty,axiom,
( ( comple9105089376463352645_set_a @ bot_bo3380559777022489994_set_a )
= top_top_set_set_a ) ).
% Inter_empty
thf(fact_1221_disjoint__iff__not__equal,axiom,
! [A3: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A3 @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1222_Int__Diff__disjoint,axiom,
! [A3: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B2 ) @ ( minus_minus_set_a @ A3 @ B2 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_1223_Int__empty__right,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_1224_Int__empty__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B2 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_1225_disjoint__iff,axiom,
! [A3: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A3 @ B2 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ~ ( member_set_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1226_disjoint__iff,axiom,
! [A3: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A3 @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ~ ( member_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1227_Int__emptyI,axiom,
! [A3: set_set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ~ ( member_set_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_set_a @ A3 @ B2 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_1228_Int__emptyI,axiom,
! [A3: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ~ ( member_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_a @ A3 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_1229_Diff__triv,axiom,
! [A3: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A3 @ B2 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A3 @ B2 )
= A3 ) ) ).
% Diff_triv
thf(fact_1230_Inter__subset,axiom,
! [A3: set_set_a,B2: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A3 )
=> ( ord_less_eq_set_a @ X7 @ B2 ) )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A3 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_1231_strict__mono__on__imp__mono__on,axiom,
! [A3: set_a,F: a > set_a] :
( ( monotone_on_a_set_a @ A3 @ ord_less_a @ ord_less_set_a @ F )
=> ( monotone_on_a_set_a @ A3 @ ord_less_eq_a @ ord_less_eq_set_a @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_1232_strict__mono__on__imp__mono__on,axiom,
! [A3: set_a,F: a > $o > a] :
( ( monotone_on_a_o_a @ A3 @ ord_less_a @ ord_less_o_a @ F )
=> ( monotone_on_a_o_a @ A3 @ ord_less_eq_a @ ord_less_eq_o_a @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_1233_strict__mono__on__imp__mono__on,axiom,
! [A3: set_a,F: a > a] :
( ( monotone_on_a_a @ A3 @ ord_less_a @ ord_less_a @ F )
=> ( monotone_on_a_a @ A3 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_1234_strict__mono__on__leD,axiom,
! [A3: set_a,F: a > set_a,X: a,Y: a] :
( ( monotone_on_a_set_a @ A3 @ ord_less_a @ ord_less_set_a @ F )
=> ( ( member_a @ X @ A3 )
=> ( ( member_a @ Y @ A3 )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_1235_strict__mono__on__leD,axiom,
! [A3: set_a,F: a > $o > a,X: a,Y: a] :
( ( monotone_on_a_o_a @ A3 @ ord_less_a @ ord_less_o_a @ F )
=> ( ( member_a @ X @ A3 )
=> ( ( member_a @ Y @ A3 )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_1236_strict__mono__on__leD,axiom,
! [A3: set_a,F: a > a,X: a,Y: a] :
( ( monotone_on_a_a @ A3 @ ord_less_a @ ord_less_a @ F )
=> ( ( member_a @ X @ A3 )
=> ( ( member_a @ Y @ A3 )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_1237_mono__on__greaterD,axiom,
! [A3: set_a,G2: a > a,X: a,Y: a] :
( ( monotone_on_a_a @ A3 @ ord_less_eq_a @ ord_less_eq_a @ G2 )
=> ( ( member_a @ X @ A3 )
=> ( ( member_a @ Y @ A3 )
=> ( ( ord_less_a @ ( G2 @ Y ) @ ( G2 @ X ) )
=> ( ord_less_a @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_1238_mono__imp__mono__on,axiom,
! [F: a > set_a,A3: set_a] :
( ( monotone_on_a_set_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_set_a @ F )
=> ( monotone_on_a_set_a @ A3 @ ord_less_eq_a @ ord_less_eq_set_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1239_mono__imp__mono__on,axiom,
! [F: a > $o > a,A3: set_a] :
( ( monotone_on_a_o_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_o_a @ F )
=> ( monotone_on_a_o_a @ A3 @ ord_less_eq_a @ ord_less_eq_o_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1240_mono__imp__mono__on,axiom,
! [F: set_a > a,A3: set_set_a] :
( ( monotone_on_set_a_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_a @ F )
=> ( monotone_on_set_a_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1241_mono__imp__mono__on,axiom,
! [F: set_a > set_a,A3: set_set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( monoto7172710143293369831_set_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1242_mono__imp__mono__on,axiom,
! [F: set_a > $o > a,A3: set_set_a] :
( ( monoto9194799565806279554_a_o_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_o_a @ F )
=> ( monoto9194799565806279554_a_o_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_o_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1243_mono__imp__mono__on,axiom,
! [F: ( $o > a ) > a,A3: set_o_a] :
( ( monotone_on_o_a_a @ top_top_set_o_a @ ord_less_eq_o_a @ ord_less_eq_a @ F )
=> ( monotone_on_o_a_a @ A3 @ ord_less_eq_o_a @ ord_less_eq_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1244_mono__imp__mono__on,axiom,
! [F: ( $o > a ) > set_a,A3: set_o_a] :
( ( monoto888385181794500650_set_a @ top_top_set_o_a @ ord_less_eq_o_a @ ord_less_eq_set_a @ F )
=> ( monoto888385181794500650_set_a @ A3 @ ord_less_eq_o_a @ ord_less_eq_set_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1245_mono__imp__mono__on,axiom,
! [F: ( $o > a ) > $o > a,A3: set_o_a] :
( ( monotone_on_o_a_o_a @ top_top_set_o_a @ ord_less_eq_o_a @ ord_less_eq_o_a @ F )
=> ( monotone_on_o_a_o_a @ A3 @ ord_less_eq_o_a @ ord_less_eq_o_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1246_mono__imp__mono__on,axiom,
! [F: a > a,A3: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( monotone_on_a_a @ A3 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_1247_monoI,axiom,
! [F: ( $o > a ) > set_a] :
( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto888385181794500650_set_a @ top_top_set_o_a @ ord_less_eq_o_a @ ord_less_eq_set_a @ F ) ) ).
% monoI
thf(fact_1248_monoI,axiom,
! [F: ( $o > a ) > $o > a] :
( ! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_o_a_o_a @ top_top_set_o_a @ ord_less_eq_o_a @ ord_less_eq_o_a @ F ) ) ).
% monoI
thf(fact_1249_monoI,axiom,
! [F: a > a] :
( ! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% monoI
thf(fact_1250_monoE,axiom,
! [F: a > a,X: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1251_monoD,axiom,
! [F: a > a,X: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1252_mono__on__subset,axiom,
! [A3: set_a,F: a > a,B2: set_a] :
( ( monotone_on_a_a @ A3 @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_set_a @ B2 @ A3 )
=> ( monotone_on_a_a @ B2 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ) ).
% mono_on_subset
thf(fact_1253_insert__Diff,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1254_Diff__insert__absorb,axiom,
! [X: a,A3: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1255_in__image__insert__iff,axiom,
! [B2: set_set_a,X: a,A3: set_a] :
( ! [C5: set_a] :
( ( member_set_a @ C5 @ B2 )
=> ~ ( member_a @ X @ C5 ) )
=> ( ( member_set_a @ A3 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B2 ) )
= ( ( member_a @ X @ A3 )
& ( member_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1256_subset__Diff__insert,axiom,
! [A3: set_a,B2: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B2 @ C2 ) )
& ~ ( member_a @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1257_mono__invE,axiom,
! [F: a > a,X: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_a @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_a @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1258_strict__mono__mono,axiom,
! [F: a > a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_a @ ord_less_a @ F )
=> ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% strict_mono_mono
thf(fact_1259_mono__strict__invE,axiom,
! [F: a > a,X: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_a @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_a @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_1260_strict__mono__less__eq,axiom,
! [F: a > a,X: a,Y: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_a @ ord_less_a @ F )
=> ( ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_a @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_1261_subset__insert__iff,axiom,
! [A3: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1262_Ioc__disjoint,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ A @ B ) @ ( set_or4472690218693186638Most_a @ C @ D ) )
= bot_bot_set_a )
= ( ( ord_less_eq_a @ B @ A )
| ( ord_less_eq_a @ D @ C )
| ( ord_less_eq_a @ B @ C )
| ( ord_less_eq_a @ D @ A ) ) ) ).
% Ioc_disjoint
thf(fact_1263_psubset__insert__iff,axiom,
! [A3: set_a,X: a,B2: set_a] :
( ( ord_less_set_a @ A3 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ B2 )
=> ( ord_less_set_a @ A3 @ B2 ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( ( member_a @ X @ A3 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1264_member__remove,axiom,
! [X: a,Y: a,A3: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A3 ) )
= ( ( member_a @ X @ A3 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1265_coinduct__set,axiom,
! [F: set_a > set_a,A: a,X6: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ A @ X6 )
=> ( ( ord_less_eq_set_a @ X6 @ ( F @ ( sup_sup_set_a @ X6 @ ( comple3341859861669737308_set_a @ F ) ) ) )
=> ( member_a @ A @ ( comple3341859861669737308_set_a @ F ) ) ) ) ) ).
% coinduct_set
thf(fact_1266_gfp__fun__UnI2,axiom,
! [F: set_a > set_a,A: a,X6: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ A @ ( comple3341859861669737308_set_a @ F ) )
=> ( member_a @ A @ ( F @ ( sup_sup_set_a @ X6 @ ( comple3341859861669737308_set_a @ F ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_1267_weak__coinduct__image,axiom,
! [A: a,X6: set_a,G2: a > a,F: set_a > set_a] :
( ( member_a @ A @ X6 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G2 @ X6 ) @ ( F @ ( image_a_a @ G2 @ X6 ) ) )
=> ( member_a @ ( G2 @ A ) @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_1268_weak__coinduct,axiom,
! [A: a,X6: set_a,F: set_a > set_a] :
( ( member_a @ A @ X6 )
=> ( ( ord_less_eq_set_a @ X6 @ ( F @ X6 ) )
=> ( member_a @ A @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct
thf(fact_1269_def__coinduct__set,axiom,
! [A3: set_a,F: set_a > set_a,A: a,X6: set_a] :
( ( A3
= ( comple3341859861669737308_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ A @ X6 )
=> ( ( ord_less_eq_set_a @ X6 @ ( F @ ( sup_sup_set_a @ X6 @ A3 ) ) )
=> ( member_a @ A @ A3 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_1270_bdd__below_OI,axiom,
! [A3: set_a,M3: a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ord_less_eq_a @ M3 @ X2 ) )
=> ( condit5901475214736682318elow_a @ A3 ) ) ).
% bdd_below.I
thf(fact_1271_bdd__belowI,axiom,
! [A3: set_a,M2: a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ord_less_eq_a @ M2 @ X2 ) )
=> ( condit5901475214736682318elow_a @ A3 ) ) ).
% bdd_belowI
thf(fact_1272_bdd__below_OI2,axiom,
! [A3: set_a,M3: a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ord_less_eq_a @ M3 @ ( F @ X2 ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A3 ) ) ) ).
% bdd_below.I2
thf(fact_1273_bdd__belowI2,axiom,
! [A3: set_a,M2: a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ord_less_eq_a @ M2 @ ( F @ X2 ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A3 ) ) ) ).
% bdd_belowI2
thf(fact_1274_bdd__below_Ounfold,axiom,
( condit5901475214736682318elow_a
= ( ^ [A2: set_a] :
? [M5: a] :
! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_a @ M5 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1275_bdd__below_OE,axiom,
! [A3: set_a] :
( ( condit5901475214736682318elow_a @ A3 )
=> ~ ! [M6: a] :
~ ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( ord_less_eq_a @ M6 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1276_bdd__below__image__mono,axiom,
! [F: a > a,A3: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( condit5901475214736682318elow_a @ A3 )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A3 ) ) ) ) ).
% bdd_below_image_mono
% Conjectures (1)
thf(conj_0,conjecture,
( ( minimum_Minimum_a @ s )
= m ) ).
%------------------------------------------------------------------------------