TPTP Problem File: SLH0066^1.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_00051_001512__28700372_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1424 ( 582 unt; 146 typ; 0 def)
% Number of atoms : 3982 (1212 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10914 ( 358 ~; 81 |; 240 &;8601 @)
% ( 0 <=>;1634 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 1672 (1672 >; 0 *; 0 +; 0 <<)
% Number of symbols : 142 ( 139 usr; 11 con; 0-4 aty)
% Number of variables : 3741 ( 273 ^;3385 !; 83 ?;3741 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:52:39.553
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__a_M_Eo_J_J_J,type,
set_a_a_o: $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 (139)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
comple4872679785363069173_a_a_o: set_a_a_o > a > a > $o ).
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_Itf__a_J,type,
comple6135023378680113637_set_a: set_set_a > set_a ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001_062_I_Eo_Mtf__a_J,type,
condit8636343168986299771ow_o_a: set_o_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Set__Oset_Itf__a_J,type,
condit8937546108433946286_set_a: set_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001tf__a,type,
condit5901475214736682318elow_a: set_a > $o ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > 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_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
inf_inf_a_a_o: ( a > a > $o ) > ( a > a > $o ) > a > a > $o ).
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_I_062_Itf__a_M_062_Itf__a_M_Eo_J_J_J,type,
inf_inf_set_a_a_o: set_a_a_o > set_a_a_o > set_a_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_Osemilattice__neutr__order_001t__Set__Oset_Itf__a_J,type,
semila2496817875450240012_set_a: ( set_a > set_a > set_a ) > set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_Itf__a_J_M_062_It__Set__Oset_Itf__a_J_M_Eo_J_J,type,
sup_su8131220024717662786et_a_o: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > set_a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
sup_sup_a_a_o: ( a > a > $o ) > ( a > a > $o ) > a > a > $o ).
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_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001tf__a,type,
lattic3288624042836100505on_a_a: ( a > a ) > set_a > 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_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
bot_bot_a_a_o: a > 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_001_Eo,type,
bot_bot_o: $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_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_062_Itf__a_M_Eo_J_J,type,
ord_less_a_a_o: ( a > a > $o ) > ( a > a > $o ) > $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_062_Itf__a_M_Eo_J_J,type,
ord_less_eq_a_a_o: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_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_001_062_I_Eo_Mtf__a_J,type,
ordering_o_a: ( ( $o > a ) > ( $o > a ) > $o ) > ( ( $o > a ) > ( $o > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Set__Oset_Itf__a_J,type,
ordering_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001tf__a,type,
ordering_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oordering__axioms_001tf__a,type,
ordering_axioms_a: ( a > a > $o ) > ( a > a > $o ) > $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_Oordering__top_001tf__a,type,
ordering_top_a: ( a > a > $o ) > ( a > a > $o ) > a > $o ).
thf(sy_c_Orderings_Oordering__top__axioms_001tf__a,type,
orderi2381714506568756172ioms_a: ( a > a > $o ) > 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_Opreordering_001_062_I_Eo_Mtf__a_J,type,
preordering_o_a: ( ( $o > a ) > ( $o > a ) > $o ) > ( ( $o > a ) > ( $o > a ) > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Set__Oset_Itf__a_J,type,
preordering_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001tf__a,type,
preordering_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Opreordering__axioms_001tf__a,type,
preordering_axioms_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
top_top_a_a_o: a > 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_001t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
top_top_set_a_o: set_a_o ).
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_Itf__a_M_062_Itf__a_M_Eo_J_J_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
image_a_a_o_a_a_o: ( ( a > a > $o ) > a > a > $o ) > set_a_a_o > set_a_a_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1042221919965026181_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_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_001t__Set__Oset_Itf__a_J,type,
remove_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__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_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_001_062_Itf__a_M_Eo_J,type,
set_ord_atLeast_a_o: ( a > $o ) > set_a_o ).
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_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_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_Zorn_Ochain__subset_001tf__a,type,
chain_subset_a: set_set_a > $o ).
thf(sy_c_Zorn_Ochains_001tf__a,type,
chains_a: set_set_a > set_set_set_a ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_Itf__a_J,type,
pred_chain_set_a: set_set_a > ( set_a > set_a > $o ) > set_set_a > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001tf__a,type,
pred_chain_a: set_a > ( a > a > $o ) > set_a > $o ).
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_062_Itf__a_M_Eo_J_J,type,
member_a_a_o: ( a > a > $o ) > set_a_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_thesis____,type,
thesis: $o ).
% Relevant facts (1276)
thf(fact_0_assms,axiom,
minimu7473987258551571531imum_a @ s ).
% assms
thf(fact_1__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,
? [X: a] :
( ( member_a @ X @ s )
& ! [Xa: a] :
( ( member_a @ Xa @ s )
=> ( ord_less_eq_a @ X @ Xa ) ) ) ).
% \<open>\<exists>x. x \<in> S \<and> Ball S ((\<le>) x)\<close>
thf(fact_2_order__refl,axiom,
! [X2: $o > a] : ( ord_less_eq_o_a @ X2 @ X2 ) ).
% order_refl
thf(fact_3_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_4_order__refl,axiom,
! [X2: a] : ( ord_less_eq_a @ X2 @ X2 ) ).
% order_refl
thf(fact_5_dual__order_Orefl,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% dual_order.refl
thf(fact_6_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_7_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_8_has__Minimum__def,axiom,
( minimu4657282916794952894um_o_a
= ( ^ [S: set_o_a] :
? [X3: $o > a] :
( ( member_o_a @ X3 @ S )
& ! [Y: $o > a] :
( ( member_o_a @ Y @ S )
=> ( ord_less_eq_o_a @ X3 @ Y ) ) ) ) ) ).
% has_Minimum_def
thf(fact_9_has__Minimum__def,axiom,
( minimu6896447672505010603_set_a
= ( ^ [S: set_set_a] :
? [X3: set_a] :
( ( member_set_a @ X3 @ S )
& ! [Y: set_a] :
( ( member_set_a @ Y @ S )
=> ( ord_less_eq_set_a @ X3 @ Y ) ) ) ) ) ).
% has_Minimum_def
thf(fact_10_has__Minimum__def,axiom,
( minimu7473987258551571531imum_a
= ( ^ [S: set_a] :
? [X3: a] :
( ( member_a @ X3 @ S )
& ! [Y: a] :
( ( member_a @ Y @ S )
=> ( ord_less_eq_a @ X3 @ Y ) ) ) ) ) ).
% has_Minimum_def
thf(fact_11_eqMinimumI,axiom,
! [X2: a,S2: set_a] :
( ( member_a @ X2 @ S2 )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ S2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) )
=> ( ( minimum_Minimum_a @ S2 )
= X2 ) ) ) ).
% eqMinimumI
thf(fact_12_has__Maximum__def,axiom,
( minimu315547183909508560um_o_a
= ( ^ [S: set_o_a] :
? [X3: $o > a] :
( ( member_o_a @ X3 @ S )
& ! [Y: $o > a] :
( ( member_o_a @ Y @ S )
=> ( ord_less_eq_o_a @ Y @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_13_has__Maximum__def,axiom,
( minimu8775777210878807577_set_a
= ( ^ [S: set_set_a] :
? [X3: set_a] :
( ( member_set_a @ X3 @ S )
& ! [Y: set_a] :
( ( member_set_a @ Y @ S )
=> ( ord_less_eq_set_a @ Y @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_14_has__Maximum__def,axiom,
( minimu6197867597544231097imum_a
= ( ^ [S: set_a] :
? [X3: a] :
( ( member_a @ X3 @ S )
& ! [Y: a] :
( ( member_a @ Y @ S )
=> ( ord_less_eq_a @ Y @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_15_eqMaximumI,axiom,
! [X2: a,S2: set_a] :
( ( member_a @ X2 @ S2 )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ S2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ( minimum_Maximum_a @ S2 )
= X2 ) ) ) ).
% eqMaximumI
thf(fact_16_verit__comp__simplify1_I2_J,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_17_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_18_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_19_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_20_le__cases3,axiom,
! [X2: a,Y3: a,Z: a] :
( ( ( ord_less_eq_a @ X2 @ Y3 )
=> ~ ( ord_less_eq_a @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_a @ Y3 @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Z ) )
=> ( ( ( ord_less_eq_a @ X2 @ Z )
=> ~ ( ord_less_eq_a @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_a @ Z @ Y3 )
=> ~ ( ord_less_eq_a @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_a @ Y3 @ Z )
=> ~ ( ord_less_eq_a @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_a @ Z @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_21_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > a,Z2: $o > a] : ( Y4 = Z2 ) )
= ( ^ [X3: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X3 @ Y )
& ( ord_less_eq_o_a @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_22_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
& ( ord_less_eq_set_a @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_23_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [X3: a,Y: a] :
( ( ord_less_eq_a @ X3 @ Y )
& ( ord_less_eq_a @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_24_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_25_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_26_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_27_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_28_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_29_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_30_has__MaximumD_I1_J,axiom,
! [S2: set_a] :
( ( minimu6197867597544231097imum_a @ S2 )
=> ( member_a @ ( minimum_Maximum_a @ S2 ) @ S2 ) ) ).
% has_MaximumD(1)
thf(fact_31_has__MaximumD_I2_J,axiom,
! [S2: set_a,X2: a] :
( ( minimu6197867597544231097imum_a @ S2 )
=> ( ( member_a @ X2 @ S2 )
=> ( ord_less_eq_a @ X2 @ ( minimum_Maximum_a @ S2 ) ) ) ) ).
% has_MaximumD(2)
thf(fact_32_order__antisym__conv,axiom,
! [Y3: $o > a,X2: $o > a] :
( ( ord_less_eq_o_a @ Y3 @ X2 )
=> ( ( ord_less_eq_o_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_33_order__antisym__conv,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_34_order__antisym__conv,axiom,
! [Y3: a,X2: a] :
( ( ord_less_eq_a @ Y3 @ X2 )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_35_linorder__le__cases,axiom,
! [X2: a,Y3: a] :
( ~ ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_36_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_37_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_38_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_39_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_40_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_41_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_42_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_43_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_44_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_45_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_46_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_47_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_48_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_49_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_50_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_51_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_52_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_53_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_54_linorder__linear,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
| ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_55_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_56_order__eq__refl,axiom,
! [X2: $o > a,Y3: $o > a] :
( ( X2 = Y3 )
=> ( ord_less_eq_o_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_57_order__eq__refl,axiom,
! [X2: set_a,Y3: set_a] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_58_order__eq__refl,axiom,
! [X2: a,Y3: a] :
( ( X2 = Y3 )
=> ( ord_less_eq_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_59_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_60_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_61_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_62_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_63_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_64_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_65_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_66_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_67_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_68_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_69_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_70_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_71_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_72_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_73_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_74_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_75_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_76_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_77_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > a,Z2: $o > a] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ A2 @ B2 )
& ( ord_less_eq_o_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_78_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_79_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_80_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_81_le__funI,axiom,
! [F: $o > a,G2: $o > a] :
( ! [X: $o] : ( ord_less_eq_a @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_less_eq_o_a @ F @ G2 ) ) ).
% le_funI
thf(fact_82_le__funE,axiom,
! [F: $o > a,G2: $o > a,X2: $o] :
( ( ord_less_eq_o_a @ F @ G2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_83_le__funD,axiom,
! [F: $o > a,G2: $o > a,X2: $o] :
( ( ord_less_eq_o_a @ F @ G2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_84_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_85_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_86_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_87_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_88_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_89_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_90_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_91_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_92_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_93_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > a,Z2: $o > a] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A2 )
& ( ord_less_eq_o_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_94_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_95_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_96_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_97_order__trans,axiom,
! [X2: $o > a,Y3: $o > a,Z: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ( ord_less_eq_o_a @ Y3 @ Z )
=> ( ord_less_eq_o_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_98_order__trans,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_99_order__trans,axiom,
! [X2: a,Y3: a,Z: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ Z )
=> ( ord_less_eq_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_100_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_101_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_102_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_103_order__antisym,axiom,
! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ( ord_less_eq_o_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_104_order__antisym,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_105_order__antisym,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_106_Greatest__equality,axiom,
! [P: ( $o > a ) > $o,X2: $o > a] :
( ( P @ X2 )
=> ( ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_o_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_107_Greatest__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_108_Greatest__equality,axiom,
! [P: a > $o,X2: a] :
( ( P @ X2 )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_109_GreatestI2__order,axiom,
! [P: ( $o > a ) > $o,X2: $o > a,Q: ( $o > a ) > $o] :
( ( P @ X2 )
=> ( ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ Y2 @ X2 ) )
=> ( ! [X: $o > a] :
( ( P @ X )
=> ( ! [Y5: $o > a] :
( ( P @ Y5 )
=> ( ord_less_eq_o_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_o_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_110_GreatestI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_111_GreatestI2__order,axiom,
! [P: a > $o,X2: a,Q: a > $o] :
( ( P @ X2 )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ! [X: a] :
( ( P @ X )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_112_Ball__def,axiom,
( ball_set_a
= ( ^ [A4: set_set_a,P2: set_a > $o] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
=> ( P2 @ X3 ) ) ) ) ).
% Ball_def
thf(fact_113_Ball__def,axiom,
( ball_a
= ( ^ [A4: set_a,P2: a > $o] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( P2 @ X3 ) ) ) ) ).
% Ball_def
thf(fact_114_ball__reg,axiom,
! [R: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ! [X: set_a] :
( ( member_set_a @ X @ R )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ R )
=> ( P @ X ) )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_115_ball__reg,axiom,
! [R: set_a,P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( member_a @ X @ R )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ R )
=> ( P @ X ) )
=> ! [X4: a] :
( ( member_a @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_116_Ball__Collect,axiom,
( ball_a
= ( ^ [A4: set_a,P2: a > $o] : ( ord_less_eq_set_a @ A4 @ ( collect_a @ P2 ) ) ) ) ).
% Ball_Collect
thf(fact_117_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_118_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_119_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_120_order_Opartial__preordering__axioms,axiom,
partia5423788306336055317ng_o_a @ ord_less_eq_o_a ).
% order.partial_preordering_axioms
thf(fact_121_order_Opartial__preordering__axioms,axiom,
partia6602192050731689876_set_a @ ord_less_eq_set_a ).
% order.partial_preordering_axioms
thf(fact_122_order_Opartial__preordering__axioms,axiom,
partia125584492769400372ring_a @ ord_less_eq_a ).
% order.partial_preordering_axioms
thf(fact_123_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_125_Collect__mem__eq,axiom,
! [A5: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_126_Collect__mem__eq,axiom,
! [A5: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_127_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_128_Least__equality,axiom,
! [P: ( $o > a ) > $o,X2: $o > a] :
( ( P @ X2 )
=> ( ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ X2 @ Y2 ) )
=> ( ( ord_Least_o_a @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_129_Least__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) )
=> ( ( ord_Least_set_a @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_130_Least__equality,axiom,
! [P: a > $o,X2: a] :
( ( P @ X2 )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) )
=> ( ( ord_Least_a @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_131_LeastI2__order,axiom,
! [P: ( $o > a ) > $o,X2: $o > a,Q: ( $o > a ) > $o] :
( ( P @ X2 )
=> ( ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ X2 @ Y2 ) )
=> ( ! [X: $o > a] :
( ( P @ X )
=> ( ! [Y5: $o > a] :
( ( P @ Y5 )
=> ( ord_less_eq_o_a @ X @ Y5 ) )
=> ( Q @ X ) ) )
=> ( Q @ ( ord_Least_o_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_132_LeastI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ X @ Y5 ) )
=> ( Q @ X ) ) )
=> ( Q @ ( ord_Least_set_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_133_LeastI2__order,axiom,
! [P: a > $o,X2: a,Q: a > $o] :
( ( P @ X2 )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) )
=> ( ! [X: a] :
( ( P @ X )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ X @ Y5 ) )
=> ( Q @ X ) ) )
=> ( Q @ ( ord_Least_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_134_Least1__le,axiom,
! [P: ( $o > a ) > $o,Z: $o > a] :
( ? [X4: $o > a] :
( ( P @ X4 )
& ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ X4 @ Y2 ) )
& ! [Y2: $o > a] :
( ( ( P @ Y2 )
& ! [Ya: $o > a] :
( ( P @ Ya )
=> ( ord_less_eq_o_a @ Y2 @ Ya ) ) )
=> ( Y2 = X4 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq_o_a @ ( ord_Least_o_a @ P ) @ Z ) ) ) ).
% Least1_le
thf(fact_135_Least1__le,axiom,
! [P: set_a > $o,Z: set_a] :
( ? [X4: set_a] :
( ( P @ X4 )
& ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ X4 @ Y2 ) )
& ! [Y2: set_a] :
( ( ( P @ Y2 )
& ! [Ya: set_a] :
( ( P @ Ya )
=> ( ord_less_eq_set_a @ Y2 @ Ya ) ) )
=> ( Y2 = X4 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq_set_a @ ( ord_Least_set_a @ P ) @ Z ) ) ) ).
% Least1_le
thf(fact_136_Least1__le,axiom,
! [P: a > $o,Z: a] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ X4 @ Y2 ) )
& ! [Y2: a] :
( ( ( P @ Y2 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( ord_less_eq_a @ Y2 @ Ya ) ) )
=> ( Y2 = X4 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq_a @ ( ord_Least_a @ P ) @ Z ) ) ) ).
% Least1_le
thf(fact_137_subsetI,axiom,
! [A5: set_set_a,B4: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( member_set_a @ X @ B4 ) )
=> ( ord_le3724670747650509150_set_a @ A5 @ B4 ) ) ).
% subsetI
thf(fact_138_subsetI,axiom,
! [A5: set_a,B4: set_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ X @ B4 ) )
=> ( ord_less_eq_set_a @ A5 @ B4 ) ) ).
% subsetI
thf(fact_139_subset__antisym,axiom,
! [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A5 )
=> ( A5 = B4 ) ) ) ).
% subset_antisym
thf(fact_140_in__mono,axiom,
! [A5: set_set_a,B4: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B4 )
=> ( ( member_set_a @ X2 @ A5 )
=> ( member_set_a @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_141_in__mono,axiom,
! [A5: set_a,B4: set_a,X2: a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_142_subsetD,axiom,
! [A5: set_set_a,B4: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B4 )
=> ( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_143_subsetD,axiom,
! [A5: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( member_a @ C @ A5 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_144_equalityE,axiom,
! [A5: set_a,B4: set_a] :
( ( A5 = B4 )
=> ~ ( ( ord_less_eq_set_a @ A5 @ B4 )
=> ~ ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ).
% equalityE
thf(fact_145_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
=> ( member_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_146_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_147_equalityD1,axiom,
! [A5: set_a,B4: set_a] :
( ( A5 = B4 )
=> ( ord_less_eq_set_a @ A5 @ B4 ) ) ).
% equalityD1
thf(fact_148_equalityD2,axiom,
! [A5: set_a,B4: set_a] :
( ( A5 = B4 )
=> ( ord_less_eq_set_a @ B4 @ A5 ) ) ).
% equalityD2
thf(fact_149_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A4 )
=> ( member_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_150_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_151_subset__refl,axiom,
! [A5: set_a] : ( ord_less_eq_set_a @ A5 @ A5 ) ).
% subset_refl
thf(fact_152_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_153_subset__trans,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A5 @ C2 ) ) ) ).
% subset_trans
thf(fact_154_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_155_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_156_partial__preordering__def,axiom,
( partia125584492769400372ring_a
= ( ^ [Less_eq: a > a > $o] :
( ! [A2: a] : ( Less_eq @ A2 @ A2 )
& ! [A2: a,B2: a,C3: a] :
( ( Less_eq @ A2 @ B2 )
=> ( ( Less_eq @ B2 @ C3 )
=> ( Less_eq @ A2 @ C3 ) ) ) ) ) ) ).
% partial_preordering_def
thf(fact_157_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_158_partial__preordering_Ointro,axiom,
! [Less_eq2: a > a > $o] :
( ! [A3: a] : ( Less_eq2 @ A3 @ A3 )
=> ( ! [A3: a,B3: a,C4: a] :
( ( Less_eq2 @ A3 @ B3 )
=> ( ( Less_eq2 @ B3 @ C4 )
=> ( Less_eq2 @ A3 @ C4 ) ) )
=> ( partia125584492769400372ring_a @ Less_eq2 ) ) ) ).
% partial_preordering.intro
thf(fact_159_partial__preordering_Orefl,axiom,
! [Less_eq2: a > a > $o,A: a] :
( ( partia125584492769400372ring_a @ Less_eq2 )
=> ( Less_eq2 @ A @ A ) ) ).
% partial_preordering.refl
thf(fact_160_Least1I,axiom,
! [P: ( $o > a ) > $o] :
( ? [X4: $o > a] :
( ( P @ X4 )
& ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ X4 @ Y2 ) )
& ! [Y2: $o > a] :
( ( ( P @ Y2 )
& ! [Ya: $o > a] :
( ( P @ Ya )
=> ( ord_less_eq_o_a @ Y2 @ Ya ) ) )
=> ( Y2 = X4 ) ) )
=> ( P @ ( ord_Least_o_a @ P ) ) ) ).
% Least1I
thf(fact_161_Least1I,axiom,
! [P: set_a > $o] :
( ? [X4: set_a] :
( ( P @ X4 )
& ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ X4 @ Y2 ) )
& ! [Y2: set_a] :
( ( ( P @ Y2 )
& ! [Ya: set_a] :
( ( P @ Ya )
=> ( ord_less_eq_set_a @ Y2 @ Ya ) ) )
=> ( Y2 = X4 ) ) )
=> ( P @ ( ord_Least_set_a @ P ) ) ) ).
% Least1I
thf(fact_162_Least1I,axiom,
! [P: a > $o] :
( ? [X4: a] :
( ( P @ X4 )
& ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ X4 @ Y2 ) )
& ! [Y2: a] :
( ( ( P @ Y2 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( ord_less_eq_a @ Y2 @ Ya ) ) )
=> ( Y2 = X4 ) ) )
=> ( P @ ( ord_Least_a @ P ) ) ) ).
% Least1I
thf(fact_163_ordering_Oaxioms_I1_J,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o] :
( ( ordering_a @ Less_eq2 @ Less )
=> ( partia125584492769400372ring_a @ Less_eq2 ) ) ).
% ordering.axioms(1)
thf(fact_164_preordering_Oaxioms_I1_J,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o] :
( ( preordering_a @ Less_eq2 @ Less )
=> ( partia125584492769400372ring_a @ Less_eq2 ) ) ).
% preordering.axioms(1)
thf(fact_165_greaterThan__subset__iff,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ X2 ) @ ( set_or8632414552788122084Than_a @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% greaterThan_subset_iff
thf(fact_166_lessThan__subset__iff,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_lessThan_a @ X2 ) @ ( set_ord_lessThan_a @ Y3 ) )
= ( ord_less_eq_a @ X2 @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_167_atLeast__subset__iff,axiom,
! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_ord_atLeast_o_a @ X2 ) @ ( set_ord_atLeast_o_a @ Y3 ) )
= ( ord_less_eq_o_a @ Y3 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_168_atLeast__subset__iff,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or8362275514725411625_set_a @ X2 ) @ ( set_or8362275514725411625_set_a @ Y3 ) )
= ( ord_less_eq_set_a @ Y3 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_169_atLeast__subset__iff,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ X2 ) @ ( set_ord_atLeast_a @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_170_atMost__subset__iff,axiom,
! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_set_o_a @ ( set_ord_atMost_o_a @ X2 ) @ ( set_ord_atMost_o_a @ Y3 ) )
= ( ord_less_eq_o_a @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_171_atMost__subset__iff,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X2 ) @ ( set_ord_atMost_set_a @ Y3 ) )
= ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_172_atMost__subset__iff,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ X2 ) @ ( set_ord_atMost_a @ Y3 ) )
= ( ord_less_eq_a @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_173_pairwise__mono,axiom,
! [P: a > a > $o,A5: set_a,Q: a > a > $o,B4: set_a] :
( ( pairwise_a @ P @ A5 )
=> ( ! [X: a,Y2: a] :
( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) )
=> ( ( ord_less_eq_set_a @ B4 @ A5 )
=> ( pairwise_a @ Q @ B4 ) ) ) ) ).
% pairwise_mono
thf(fact_174_pairwise__subset,axiom,
! [P: a > a > $o,S2: set_a,T2: set_a] :
( ( pairwise_a @ P @ S2 )
=> ( ( ord_less_eq_set_a @ T2 @ S2 )
=> ( pairwise_a @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_175_ball__empty,axiom,
! [P: a > $o,X4: a] :
( ( member_a @ X4 @ bot_bot_set_a )
=> ( P @ X4 ) ) ).
% ball_empty
thf(fact_176_subset__empty,axiom,
! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ bot_bot_set_a )
= ( A5 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_177_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_178_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_179_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_180_all__not__in__conv,axiom,
! [A5: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A5 ) )
= ( A5 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_181_all__not__in__conv,axiom,
! [A5: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A5 ) )
= ( A5 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_182_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_183_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_184_atMost__eq__iff,axiom,
! [X2: a,Y3: a] :
( ( ( set_ord_atMost_a @ X2 )
= ( set_ord_atMost_a @ Y3 ) )
= ( X2 = Y3 ) ) ).
% atMost_eq_iff
thf(fact_185_lessThan__eq__iff,axiom,
! [X2: a,Y3: a] :
( ( ( set_ord_lessThan_a @ X2 )
= ( set_ord_lessThan_a @ Y3 ) )
= ( X2 = Y3 ) ) ).
% lessThan_eq_iff
thf(fact_186_atLeast__eq__iff,axiom,
! [X2: a,Y3: a] :
( ( ( set_ord_atLeast_a @ X2 )
= ( set_ord_atLeast_a @ Y3 ) )
= ( X2 = Y3 ) ) ).
% atLeast_eq_iff
thf(fact_187_greaterThan__eq__iff,axiom,
! [X2: a,Y3: a] :
( ( ( set_or8632414552788122084Than_a @ X2 )
= ( set_or8632414552788122084Than_a @ Y3 ) )
= ( X2 = Y3 ) ) ).
% greaterThan_eq_iff
thf(fact_188_empty__subsetI,axiom,
! [A5: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A5 ) ).
% empty_subsetI
thf(fact_189_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_190_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_191_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_192_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_193_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_194_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_195_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_196_pairwise__empty,axiom,
! [P: a > a > $o] : ( pairwise_a @ P @ bot_bot_set_a ) ).
% pairwise_empty
thf(fact_197_pairwise__def,axiom,
( pairwise_a
= ( ^ [R2: a > a > $o,S: set_a] :
! [X3: a] :
( ( member_a @ X3 @ S )
=> ! [Y: a] :
( ( member_a @ Y @ S )
=> ( ( X3 != Y )
=> ( R2 @ X3 @ Y ) ) ) ) ) ) ).
% pairwise_def
thf(fact_198_ex__in__conv,axiom,
! [A5: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A5 ) )
= ( A5 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_199_ex__in__conv,axiom,
! [A5: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A5 ) )
= ( A5 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_200_pairwiseI,axiom,
! [S2: set_set_a,R: set_a > set_a > $o] :
( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ S2 )
=> ( ( member_set_a @ Y2 @ S2 )
=> ( ( X != Y2 )
=> ( R @ X @ Y2 ) ) ) )
=> ( pairwise_set_a @ R @ S2 ) ) ).
% pairwiseI
thf(fact_201_pairwiseI,axiom,
! [S2: set_a,R: a > a > $o] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ S2 )
=> ( ( member_a @ Y2 @ S2 )
=> ( ( X != Y2 )
=> ( R @ X @ Y2 ) ) ) )
=> ( pairwise_a @ R @ S2 ) ) ).
% pairwiseI
thf(fact_202_pairwiseD,axiom,
! [R: set_a > set_a > $o,S2: set_set_a,X2: set_a,Y3: set_a] :
( ( pairwise_set_a @ R @ S2 )
=> ( ( member_set_a @ X2 @ S2 )
=> ( ( member_set_a @ Y3 @ S2 )
=> ( ( X2 != Y3 )
=> ( R @ X2 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_203_pairwiseD,axiom,
! [R: a > a > $o,S2: set_a,X2: a,Y3: a] :
( ( pairwise_a @ R @ S2 )
=> ( ( member_a @ X2 @ S2 )
=> ( ( member_a @ Y3 @ S2 )
=> ( ( X2 != Y3 )
=> ( R @ X2 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_204_equals0I,axiom,
! [A5: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A5 )
=> ( A5 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_205_equals0I,axiom,
! [A5: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A5 )
=> ( A5 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_206_equals0D,axiom,
! [A5: set_set_a,A: set_a] :
( ( A5 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A5 ) ) ).
% equals0D
thf(fact_207_equals0D,axiom,
! [A5: set_a,A: a] :
( ( A5 = bot_bot_set_a )
=> ~ ( member_a @ A @ A5 ) ) ).
% equals0D
thf(fact_208_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_209_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_210_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_211_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_212_preordering__strictI,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o] :
( ! [A3: a,B3: a] :
( ( Less_eq2 @ A3 @ B3 )
= ( ( Less @ A3 @ B3 )
| ( A3 = B3 ) ) )
=> ( ! [A3: a,B3: a] :
( ( Less @ A3 @ B3 )
=> ~ ( Less @ B3 @ A3 ) )
=> ( ! [A3: a] :
~ ( Less @ A3 @ A3 )
=> ( ! [A3: a,B3: a,C4: a] :
( ( Less @ A3 @ B3 )
=> ( ( Less @ B3 @ C4 )
=> ( Less @ A3 @ C4 ) ) )
=> ( preordering_a @ Less_eq2 @ Less ) ) ) ) ) ).
% preordering_strictI
thf(fact_213_ordering__strictI,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o] :
( ! [A3: a,B3: a] :
( ( Less_eq2 @ A3 @ B3 )
= ( ( Less @ A3 @ B3 )
| ( A3 = B3 ) ) )
=> ( ! [A3: a,B3: a] :
( ( Less @ A3 @ B3 )
=> ~ ( Less @ B3 @ A3 ) )
=> ( ! [A3: a] :
~ ( Less @ A3 @ A3 )
=> ( ! [A3: a,B3: a,C4: a] :
( ( Less @ A3 @ B3 )
=> ( ( Less @ B3 @ C4 )
=> ( Less @ A3 @ C4 ) ) )
=> ( ordering_a @ Less_eq2 @ Less ) ) ) ) ) ).
% ordering_strictI
thf(fact_214_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_215_ordering_Onot__eq__order__implies__strict,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq2 @ Less )
=> ( ( A != B )
=> ( ( Less_eq2 @ A @ B )
=> ( Less @ A @ B ) ) ) ) ).
% ordering.not_eq_order_implies_strict
thf(fact_216_preordering_Ostrict__implies__order,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( preordering_a @ Less_eq2 @ Less )
=> ( ( Less @ A @ B )
=> ( Less_eq2 @ A @ B ) ) ) ).
% preordering.strict_implies_order
thf(fact_217_ordering_Ostrict__implies__not__eq,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq2 @ Less )
=> ( ( Less @ A @ B )
=> ( A != B ) ) ) ).
% ordering.strict_implies_not_eq
thf(fact_218_preordering_Ostrict__iff__not,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( preordering_a @ Less_eq2 @ Less )
=> ( ( Less @ A @ B )
= ( ( Less_eq2 @ A @ B )
& ~ ( Less_eq2 @ B @ A ) ) ) ) ).
% preordering.strict_iff_not
thf(fact_219_preordering_Ostrict__trans2,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a,C: a] :
( ( preordering_a @ Less_eq2 @ Less )
=> ( ( Less @ A @ B )
=> ( ( Less_eq2 @ B @ C )
=> ( Less @ A @ C ) ) ) ) ).
% preordering.strict_trans2
thf(fact_220_preordering_Ostrict__trans1,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a,C: a] :
( ( preordering_a @ Less_eq2 @ Less )
=> ( ( Less_eq2 @ A @ B )
=> ( ( Less @ B @ C )
=> ( Less @ A @ C ) ) ) ) ).
% preordering.strict_trans1
thf(fact_221_ordering_Ostrict__iff__order,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq2 @ Less )
=> ( ( Less @ A @ B )
= ( ( Less_eq2 @ A @ B )
& ( A != B ) ) ) ) ).
% ordering.strict_iff_order
thf(fact_222_ordering_Oorder__iff__strict,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq2 @ Less )
=> ( ( Less_eq2 @ A @ B )
= ( ( Less @ A @ B )
| ( A = B ) ) ) ) ).
% ordering.order_iff_strict
thf(fact_223_preordering_Ostrict__trans,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a,C: a] :
( ( preordering_a @ Less_eq2 @ Less )
=> ( ( Less @ A @ B )
=> ( ( Less @ B @ C )
=> ( Less @ A @ C ) ) ) ) ).
% preordering.strict_trans
thf(fact_224_preordering_Oirrefl,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a] :
( ( preordering_a @ Less_eq2 @ Less )
=> ~ ( Less @ A @ A ) ) ).
% preordering.irrefl
thf(fact_225_preordering_Oasym,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( preordering_a @ Less_eq2 @ Less )
=> ( ( Less @ A @ B )
=> ~ ( Less @ B @ A ) ) ) ).
% preordering.asym
thf(fact_226_ordering_Oantisym,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq2 @ Less )
=> ( ( Less_eq2 @ A @ B )
=> ( ( Less_eq2 @ B @ A )
=> ( A = B ) ) ) ) ).
% ordering.antisym
thf(fact_227_ordering_Oeq__iff,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A: a,B: a] :
( ( ordering_a @ Less_eq2 @ Less )
=> ( ( A = B )
= ( ( Less_eq2 @ A @ B )
& ( Less_eq2 @ B @ A ) ) ) ) ).
% ordering.eq_iff
thf(fact_228_bot_Oextremum__uniqueI,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
=> ( A = bot_bot_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_229_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_230_bot_Oextremum__unique,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
= ( A = bot_bot_a_o ) ) ).
% bot.extremum_unique
thf(fact_231_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_232_bot_Oextremum,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A ) ).
% bot.extremum
thf(fact_233_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_234_subset__emptyI,axiom,
! [A5: set_set_a] :
( ! [X: set_a] :
~ ( member_set_a @ X @ A5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_235_subset__emptyI,axiom,
! [A5: set_a] :
( ! [X: a] :
~ ( member_a @ X @ A5 )
=> ( ord_less_eq_set_a @ A5 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_236_ordering__def,axiom,
( ordering_a
= ( ^ [Less_eq: a > a > $o,Less2: a > a > $o] :
( ( partia125584492769400372ring_a @ Less_eq )
& ( ordering_axioms_a @ Less_eq @ Less2 ) ) ) ) ).
% ordering_def
thf(fact_237_ordering_Ointro,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o] :
( ( partia125584492769400372ring_a @ Less_eq2 )
=> ( ( ordering_axioms_a @ Less_eq2 @ Less )
=> ( ordering_a @ Less_eq2 @ Less ) ) ) ).
% ordering.intro
thf(fact_238_preordering__def,axiom,
( preordering_a
= ( ^ [Less_eq: a > a > $o,Less2: a > a > $o] :
( ( partia125584492769400372ring_a @ Less_eq )
& ( preordering_axioms_a @ Less_eq @ Less2 ) ) ) ) ).
% preordering_def
thf(fact_239_preordering_Ointro,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o] :
( ( partia125584492769400372ring_a @ Less_eq2 )
=> ( ( preordering_axioms_a @ Less_eq2 @ Less )
=> ( preordering_a @ Less_eq2 @ Less ) ) ) ).
% preordering.intro
thf(fact_240_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A4: set_a] : ( A4 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_241_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_242_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_243_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_244_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_245_ordering_Oaxioms_I2_J,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o] :
( ( ordering_a @ Less_eq2 @ Less )
=> ( ordering_axioms_a @ Less_eq2 @ Less ) ) ).
% ordering.axioms(2)
thf(fact_246_preordering_Oaxioms_I2_J,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o] :
( ( preordering_a @ Less_eq2 @ Less )
=> ( preordering_axioms_a @ Less_eq2 @ Less ) ) ).
% preordering.axioms(2)
thf(fact_247_psubsetI,axiom,
! [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( A5 != B4 )
=> ( ord_less_set_a @ A5 @ B4 ) ) ) ).
% psubsetI
thf(fact_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_lessThan__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_ord_lessThan_a @ K ) )
= ( ord_less_a @ I @ K ) ) ).
% lessThan_iff
thf(fact_256_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_257_greaterThan__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_or8632414552788122084Than_a @ K ) )
= ( ord_less_a @ K @ I ) ) ).
% greaterThan_iff
thf(fact_258_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_259_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_260_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_261_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_262_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_263_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_264_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_265_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_266_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_267_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_268_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_269_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_270_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_271_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_272_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_273_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_274_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_275_verit__comp__simplify1_I1_J,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_276_verit__comp__simplify1_I1_J,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_277_less__imp__neq,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_278_less__imp__neq,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_279_order_Oasym,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order.asym
thf(fact_280_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_281_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_282_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_283_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_284_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_285_antisym__conv3,axiom,
! [Y3: a,X2: a] :
( ~ ( ord_less_a @ Y3 @ X2 )
=> ( ( ~ ( ord_less_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_286_linorder__cases,axiom,
! [X2: a,Y3: a] :
( ~ ( ord_less_a @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_a @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_287_dual__order_Oasym,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ~ ( ord_less_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_288_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_289_dual__order_Oirrefl,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_290_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_291_linorder__less__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A3: a,B3: a] :
( ( ord_less_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a] : ( P @ A3 @ A3 )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_292_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_293_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_294_not__less__iff__gr__or__eq,axiom,
! [X2: a,Y3: a] :
( ( ~ ( ord_less_a @ X2 @ Y3 ) )
= ( ( ord_less_a @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_295_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_296_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_297_order_Ostrict__implies__not__eq,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_298_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_299_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_300_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_301_linorder__neqE,axiom,
! [X2: a,Y3: a] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_a @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_302_order__less__asym,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_303_order__less__asym,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_304_linorder__neq__iff,axiom,
! [X2: a,Y3: a] :
( ( X2 != Y3 )
= ( ( ord_less_a @ X2 @ Y3 )
| ( ord_less_a @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_305_order__less__asym_H,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_306_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_307_order__less__trans,axiom,
! [X2: a,Y3: a,Z: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ( ord_less_a @ Y3 @ Z )
=> ( ord_less_a @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_308_order__less__trans,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_309_ord__eq__less__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_310_ord__eq__less__subst,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_311_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_312_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_313_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_314_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_315_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_316_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_317_order__less__irrefl,axiom,
! [X2: a] :
~ ( ord_less_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_318_order__less__irrefl,axiom,
! [X2: set_a] :
~ ( ord_less_set_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_319_order__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_320_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_321_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_322_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_323_order__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_324_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_325_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_326_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_327_order__less__not__sym,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_328_order__less__not__sym,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_329_order__less__imp__triv,axiom,
! [X2: a,Y3: a,P: $o] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ( ord_less_a @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_330_order__less__imp__triv,axiom,
! [X2: set_a,Y3: set_a,P: $o] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_331_linorder__less__linear,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_a @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_332_order__less__imp__not__eq,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_333_order__less__imp__not__eq,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_334_order__less__imp__not__eq2,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_335_order__less__imp__not__eq2,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_336_order__less__imp__not__less,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_337_order__less__imp__not__less,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_338_lessThan__strict__subset__iff,axiom,
! [M: a,N: a] :
( ( ord_less_set_a @ ( set_ord_lessThan_a @ M ) @ ( set_ord_lessThan_a @ N ) )
= ( ord_less_a @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_339_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_340_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_341_leD,axiom,
! [Y3: $o > a,X2: $o > a] :
( ( ord_less_eq_o_a @ Y3 @ X2 )
=> ~ ( ord_less_o_a @ X2 @ Y3 ) ) ).
% leD
thf(fact_342_leD,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ~ ( ord_less_set_a @ X2 @ Y3 ) ) ).
% leD
thf(fact_343_leD,axiom,
! [Y3: a,X2: a] :
( ( ord_less_eq_a @ Y3 @ X2 )
=> ~ ( ord_less_a @ X2 @ Y3 ) ) ).
% leD
thf(fact_344_leI,axiom,
! [X2: a,Y3: a] :
( ~ ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% leI
thf(fact_345_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_346_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_347_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_348_antisym__conv1,axiom,
! [X2: $o > a,Y3: $o > a] :
( ~ ( ord_less_o_a @ X2 @ Y3 )
=> ( ( ord_less_eq_o_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_349_antisym__conv1,axiom,
! [X2: set_a,Y3: set_a] :
( ~ ( ord_less_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_350_antisym__conv1,axiom,
! [X2: a,Y3: a] :
( ~ ( ord_less_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_351_antisym__conv2,axiom,
! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ( ~ ( ord_less_o_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_352_antisym__conv2,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_353_antisym__conv2,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ~ ( ord_less_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_354_less__le__not__le,axiom,
( ord_less_o_a
= ( ^ [X3: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X3 @ Y )
& ~ ( ord_less_eq_o_a @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_355_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
& ~ ( ord_less_eq_set_a @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_356_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y: a] :
( ( ord_less_eq_a @ X3 @ Y )
& ~ ( ord_less_eq_a @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_357_not__le__imp__less,axiom,
! [Y3: a,X2: a] :
( ~ ( ord_less_eq_a @ Y3 @ X2 )
=> ( ord_less_a @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_358_order_Oorder__iff__strict,axiom,
( ord_less_eq_o_a
= ( ^ [A2: $o > a,B2: $o > a] :
( ( ord_less_o_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_359_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_360_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_361_order_Ostrict__iff__order,axiom,
( ord_less_o_a
= ( ^ [A2: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_362_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_363_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_364_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_365_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_366_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_367_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_368_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_369_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_370_order_Ostrict__iff__not,axiom,
( ord_less_o_a
= ( ^ [A2: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ A2 @ B2 )
& ~ ( ord_less_eq_o_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_371_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_372_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ~ ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_373_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_o_a
= ( ^ [B2: $o > a,A2: $o > a] :
( ( ord_less_o_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_374_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_375_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_376_dual__order_Ostrict__iff__order,axiom,
( ord_less_o_a
= ( ^ [B2: $o > a,A2: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_377_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_378_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_379_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_380_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_381_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_382_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_383_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_384_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_385_dual__order_Ostrict__iff__not,axiom,
( ord_less_o_a
= ( ^ [B2: $o > a,A2: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A2 )
& ~ ( ord_less_eq_o_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_386_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_387_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ~ ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_388_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_389_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_390_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_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: 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_393_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_394_order__le__less,axiom,
( ord_less_eq_o_a
= ( ^ [X3: $o > a,Y: $o > a] :
( ( ord_less_o_a @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_395_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_396_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X3: a,Y: a] :
( ( ord_less_a @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_397_order__less__le,axiom,
( ord_less_o_a
= ( ^ [X3: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_398_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_399_order__less__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y: a] :
( ( ord_less_eq_a @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_400_linorder__not__le,axiom,
! [X2: a,Y3: a] :
( ( ~ ( ord_less_eq_a @ X2 @ Y3 ) )
= ( ord_less_a @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_401_linorder__not__less,axiom,
! [X2: a,Y3: a] :
( ( ~ ( ord_less_a @ X2 @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_402_order__less__imp__le,axiom,
! [X2: $o > a,Y3: $o > a] :
( ( ord_less_o_a @ X2 @ Y3 )
=> ( ord_less_eq_o_a @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_403_order__less__imp__le,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_404_order__less__imp__le,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_405_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_406_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_407_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_408_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_409_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_410_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_411_order__le__less__trans,axiom,
! [X2: $o > a,Y3: $o > a,Z: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ( ord_less_o_a @ Y3 @ Z )
=> ( ord_less_o_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_412_order__le__less__trans,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_413_order__le__less__trans,axiom,
! [X2: a,Y3: a,Z: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_a @ Y3 @ Z )
=> ( ord_less_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_414_order__less__le__trans,axiom,
! [X2: $o > a,Y3: $o > a,Z: $o > a] :
( ( ord_less_o_a @ X2 @ Y3 )
=> ( ( ord_less_eq_o_a @ Y3 @ Z )
=> ( ord_less_o_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_415_order__less__le__trans,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_416_order__less__le__trans,axiom,
! [X2: a,Y3: a,Z: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ Z )
=> ( ord_less_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_417_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_418_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_419_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_420_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_421_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_422_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_423_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_424_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_425_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_426_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_427_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_428_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_429_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_430_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_431_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_432_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_433_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_434_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_435_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_436_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 )
=> ( ! [X: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_437_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_438_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_439_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_440_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_441_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_442_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_443_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_444_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_o_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_445_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 )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_446_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 )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_447_linorder__le__less__linear,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
| ( ord_less_a @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_448_order__le__imp__less__or__eq,axiom,
! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ( ord_less_o_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_449_order__le__imp__less__or__eq,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_set_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_450_order__le__imp__less__or__eq,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_451_bot_Oextremum__strict,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ A @ bot_bot_a_o ) ).
% bot.extremum_strict
thf(fact_452_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_453_bot_Onot__eq__extremum,axiom,
! [A: a > $o] :
( ( A != bot_bot_a_o )
= ( ord_less_a_o @ bot_bot_a_o @ A ) ) ).
% bot.not_eq_extremum
thf(fact_454_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_455_not__psubset__empty,axiom,
! [A5: set_a] :
~ ( ord_less_set_a @ A5 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_456_psubsetE,axiom,
! [A5: set_a,B4: set_a] :
( ( ord_less_set_a @ A5 @ B4 )
=> ~ ( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ).
% psubsetE
thf(fact_457_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_458_psubset__imp__subset,axiom,
! [A5: set_a,B4: set_a] :
( ( ord_less_set_a @ A5 @ B4 )
=> ( ord_less_eq_set_a @ A5 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_459_psubset__subset__trans,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A5 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A5 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_460_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_461_subset__psubset__trans,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A5 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_462_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_set_a @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_463_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_464_order_Opreordering__axioms,axiom,
preordering_o_a @ ord_less_eq_o_a @ ord_less_o_a ).
% order.preordering_axioms
thf(fact_465_order_Opreordering__axioms,axiom,
preordering_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% order.preordering_axioms
thf(fact_466_order_Opreordering__axioms,axiom,
preordering_a @ ord_less_eq_a @ ord_less_a ).
% order.preordering_axioms
thf(fact_467_order_Oordering__axioms,axiom,
ordering_o_a @ ord_less_eq_o_a @ ord_less_o_a ).
% order.ordering_axioms
thf(fact_468_order_Oordering__axioms,axiom,
ordering_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% order.ordering_axioms
thf(fact_469_order_Oordering__axioms,axiom,
ordering_a @ ord_less_eq_a @ ord_less_a ).
% order.ordering_axioms
thf(fact_470_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_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_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_476_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_477_Un__iff,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B4 ) )
= ( ( member_set_a @ C @ A5 )
| ( member_set_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_478_Un__iff,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A5 @ B4 ) )
= ( ( member_a @ C @ A5 )
| ( member_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_479_UnCI,axiom,
! [C: set_a,B4: set_set_a,A5: set_set_a] :
( ( ~ ( member_set_a @ C @ B4 )
=> ( member_set_a @ C @ A5 ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B4 ) ) ) ).
% UnCI
thf(fact_480_UnCI,axiom,
! [C: a,B4: set_a,A5: set_a] :
( ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ A5 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B4 ) ) ) ).
% UnCI
thf(fact_481_Un__empty,axiom,
! [A5: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A5 @ B4 )
= bot_bot_set_a )
= ( ( A5 = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_482_Un__subset__iff,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A5 @ B4 ) @ C2 )
= ( ( ord_less_eq_set_a @ A5 @ C2 )
& ( ord_less_eq_set_a @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_483_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_484_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_485_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_486_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_487_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_488_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_489_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_490_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_491_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_492_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_493_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_494_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_495_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_496_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_497_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_498_ivl__disj__un__two_I2_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_eq_a @ L @ M )
=> ( ( ord_less_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M ) @ ( set_or5939364468397584554Than_a @ M @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_499_psubsetD,axiom,
! [A5: set_set_a,B4: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A5 @ B4 )
=> ( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_500_psubsetD,axiom,
! [A5: set_a,B4: set_a,C: a] :
( ( ord_less_set_a @ A5 @ B4 )
=> ( ( member_a @ C @ A5 )
=> ( member_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_501_psubset__trans,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A5 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A5 @ C2 ) ) ) ).
% psubset_trans
thf(fact_502_Un__left__commute,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ A5 @ ( sup_sup_set_a @ B4 @ C2 ) )
= ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A5 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_503_Un__left__absorb,axiom,
! [A5: set_a,B4: set_a] :
( ( sup_sup_set_a @ A5 @ ( sup_sup_set_a @ A5 @ B4 ) )
= ( sup_sup_set_a @ A5 @ B4 ) ) ).
% Un_left_absorb
thf(fact_504_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_505_Un__absorb,axiom,
! [A5: set_a] :
( ( sup_sup_set_a @ A5 @ A5 )
= A5 ) ).
% Un_absorb
thf(fact_506_Un__assoc,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A5 @ B4 ) @ C2 )
= ( sup_sup_set_a @ A5 @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_507_ball__Un,axiom,
! [A5: set_a,B4: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A5 @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( P @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_508_bex__Un,axiom,
! [A5: set_a,B4: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A5 @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A5 )
& ( P @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_509_UnI2,axiom,
! [C: set_a,B4: set_set_a,A5: set_set_a] :
( ( member_set_a @ C @ B4 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B4 ) ) ) ).
% UnI2
thf(fact_510_UnI2,axiom,
! [C: a,B4: set_a,A5: set_a] :
( ( member_a @ C @ B4 )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B4 ) ) ) ).
% UnI2
thf(fact_511_UnI1,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B4 ) ) ) ).
% UnI1
thf(fact_512_UnI1,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ A5 )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B4 ) ) ) ).
% UnI1
thf(fact_513_UnE,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B4 ) )
=> ( ~ ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_514_UnE,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A5 @ B4 ) )
=> ( ~ ( member_a @ C @ A5 )
=> ( member_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_515_ivl__disj__un__two_I6_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_eq_a @ L @ M )
=> ( ( ord_less_eq_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M ) @ ( set_or4472690218693186638Most_a @ M @ U ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_516_ivl__disj__un__two_I5_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_a @ L @ M )
=> ( ( ord_less_eq_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M ) @ ( set_or672772299803893939Most_a @ M @ U ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_517_ivl__disj__un__two_I8_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_eq_a @ L @ M )
=> ( ( ord_less_eq_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M ) @ ( set_or4472690218693186638Most_a @ M @ U ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_518_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_519_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_520_Un__empty__right,axiom,
! [A5: set_a] :
( ( sup_sup_set_a @ A5 @ bot_bot_set_a )
= A5 ) ).
% Un_empty_right
thf(fact_521_Un__empty__left,axiom,
! [B4: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_522_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( sup_sup_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_523_subset__UnE,axiom,
! [C2: set_a,A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A5 @ B4 ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A5 )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B4 )
=> ( C2
!= ( sup_sup_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_524_Un__absorb2,axiom,
! [B4: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B4 @ A5 )
=> ( ( sup_sup_set_a @ A5 @ B4 )
= A5 ) ) ).
% Un_absorb2
thf(fact_525_Un__absorb1,axiom,
! [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( sup_sup_set_a @ A5 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_526_Un__upper2,axiom,
! [B4: set_a,A5: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A5 @ B4 ) ) ).
% Un_upper2
thf(fact_527_Un__upper1,axiom,
! [A5: set_a,B4: set_a] : ( ord_less_eq_set_a @ A5 @ ( sup_sup_set_a @ A5 @ B4 ) ) ).
% Un_upper1
thf(fact_528_Un__least,axiom,
! [A5: set_a,C2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A5 @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_529_Un__mono,axiom,
! [A5: set_a,C2: set_a,B4: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A5 @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A5 @ B4 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_530_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_531_ivl__disj__un__two__touch_I3_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_a @ L @ M )
=> ( ( ord_less_eq_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M ) @ ( set_or672772299803893939Most_a @ M @ U ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_532_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_533_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_534_ivl__disj__un__two__touch_I4_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_eq_a @ L @ M )
=> ( ( ord_less_eq_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M ) @ ( set_or672772299803893939Most_a @ M @ U ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_535_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_536_minf_I7_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ~ ( ord_less_a @ T3 @ X4 ) ) ).
% minf(7)
thf(fact_537_minf_I5_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ord_less_a @ X4 @ T3 ) ) ).
% minf(5)
thf(fact_538_minf_I4_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( X4 != T3 ) ) ).
% minf(4)
thf(fact_539_minf_I3_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( X4 != T3 ) ) ).
% minf(3)
thf(fact_540_minf_I2_J,axiom,
! [P: a > $o,P3: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ X @ Z4 )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P3 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_541_minf_I1_J,axiom,
! [P: a > $o,P3: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ X @ Z4 )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P3 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_542_pinf_I7_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ord_less_a @ T3 @ X4 ) ) ).
% pinf(7)
thf(fact_543_pinf_I5_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ~ ( ord_less_a @ X4 @ T3 ) ) ).
% pinf(5)
thf(fact_544_pinf_I4_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(4)
thf(fact_545_pinf_I3_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(3)
thf(fact_546_pinf_I2_J,axiom,
! [P: a > $o,P3: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ Z4 @ X )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P3 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_547_pinf_I1_J,axiom,
! [P: a > $o,P3: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ Z4 @ X )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P3 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_548_sup__bot__left,axiom,
! [X2: a > a > $o] :
( ( sup_sup_a_a_o @ bot_bot_a_a_o @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_549_sup__bot__left,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_550_sup__bot__left,axiom,
! [X2: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_551_sup__bot__right,axiom,
! [X2: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ bot_bot_a_a_o )
= X2 ) ).
% sup_bot_right
thf(fact_552_sup__bot__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
= X2 ) ).
% sup_bot_right
thf(fact_553_sup__bot__right,axiom,
! [X2: a > $o] :
( ( sup_sup_a_o @ X2 @ bot_bot_a_o )
= X2 ) ).
% sup_bot_right
thf(fact_554_bot__eq__sup__iff,axiom,
! [X2: a > a > $o,Y3: a > a > $o] :
( ( bot_bot_a_a_o
= ( sup_sup_a_a_o @ X2 @ Y3 ) )
= ( ( X2 = bot_bot_a_a_o )
& ( Y3 = bot_bot_a_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_555_bot__eq__sup__iff,axiom,
! [X2: set_a,Y3: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X2 @ Y3 ) )
= ( ( X2 = bot_bot_set_a )
& ( Y3 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_556_bot__eq__sup__iff,axiom,
! [X2: a > $o,Y3: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ X2 @ Y3 ) )
= ( ( X2 = bot_bot_a_o )
& ( Y3 = bot_bot_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_557_sup__eq__bot__iff,axiom,
! [X2: a > a > $o,Y3: a > a > $o] :
( ( ( sup_sup_a_a_o @ X2 @ Y3 )
= bot_bot_a_a_o )
= ( ( X2 = bot_bot_a_a_o )
& ( Y3 = bot_bot_a_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_558_sup__eq__bot__iff,axiom,
! [X2: set_a,Y3: set_a] :
( ( ( sup_sup_set_a @ X2 @ Y3 )
= bot_bot_set_a )
= ( ( X2 = bot_bot_set_a )
& ( Y3 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_559_sup__eq__bot__iff,axiom,
! [X2: a > $o,Y3: a > $o] :
( ( ( sup_sup_a_o @ X2 @ Y3 )
= bot_bot_a_o )
= ( ( X2 = bot_bot_a_o )
& ( Y3 = bot_bot_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_560_sup__bot_Oeq__neutr__iff,axiom,
! [A: a > a > $o,B: a > a > $o] :
( ( ( sup_sup_a_a_o @ A @ B )
= bot_bot_a_a_o )
= ( ( A = bot_bot_a_a_o )
& ( B = bot_bot_a_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_561_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_562_sup__bot_Oeq__neutr__iff,axiom,
! [A: a > $o,B: a > $o] :
( ( ( sup_sup_a_o @ A @ B )
= bot_bot_a_o )
= ( ( A = bot_bot_a_o )
& ( B = bot_bot_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_563_sup__bot_Oleft__neutral,axiom,
! [A: a > a > $o] :
( ( sup_sup_a_a_o @ bot_bot_a_a_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_564_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_565_sup__bot_Oleft__neutral,axiom,
! [A: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_566_sup__bot_Oneutr__eq__iff,axiom,
! [A: a > a > $o,B: a > a > $o] :
( ( bot_bot_a_a_o
= ( sup_sup_a_a_o @ A @ B ) )
= ( ( A = bot_bot_a_a_o )
& ( B = bot_bot_a_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_567_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_568_sup__bot_Oneutr__eq__iff,axiom,
! [A: a > $o,B: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ A @ B ) )
= ( ( A = bot_bot_a_o )
& ( B = bot_bot_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_569_sup__apply,axiom,
( sup_sup_a_a_o
= ( ^ [F2: a > a > $o,G: a > a > $o,X3: a] : ( sup_sup_a_o @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% sup_apply
thf(fact_570_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_571_sup_Oright__idem,axiom,
! [A: a > a > $o,B: a > a > $o] :
( ( sup_sup_a_a_o @ ( sup_sup_a_a_o @ A @ B ) @ B )
= ( sup_sup_a_a_o @ A @ B ) ) ).
% sup.right_idem
thf(fact_572_sup__left__idem,axiom,
! [X2: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) )
= ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% sup_left_idem
thf(fact_573_sup__left__idem,axiom,
! [X2: a > a > $o,Y3: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ ( sup_sup_a_a_o @ X2 @ Y3 ) )
= ( sup_sup_a_a_o @ X2 @ Y3 ) ) ).
% sup_left_idem
thf(fact_574_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_575_sup_Oleft__idem,axiom,
! [A: a > a > $o,B: a > a > $o] :
( ( sup_sup_a_a_o @ A @ ( sup_sup_a_a_o @ A @ B ) )
= ( sup_sup_a_a_o @ A @ B ) ) ).
% sup.left_idem
thf(fact_576_sup__idem,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_577_sup__idem,axiom,
! [X2: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_578_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_579_sup_Oidem,axiom,
! [A: a > a > $o] :
( ( sup_sup_a_a_o @ A @ A )
= A ) ).
% sup.idem
thf(fact_580_le__sup__iff,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ X2 @ Y3 ) @ Z )
= ( ( ord_less_eq_a_a_o @ X2 @ Z )
& ( ord_less_eq_a_a_o @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_581_le__sup__iff,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ Z )
= ( ( ord_less_eq_set_a @ X2 @ Z )
& ( ord_less_eq_set_a @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_582_sup_Obounded__iff,axiom,
! [B: a > a > $o,C: a > a > $o,A: a > a > $o] :
( ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ B @ C ) @ A )
= ( ( ord_less_eq_a_a_o @ B @ A )
& ( ord_less_eq_a_a_o @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_583_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_584_sup__bot_Oright__neutral,axiom,
! [A: a > a > $o] :
( ( sup_sup_a_a_o @ A @ bot_bot_a_a_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_585_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_586_sup__bot_Oright__neutral,axiom,
! [A: a > $o] :
( ( sup_sup_a_o @ A @ bot_bot_a_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_587_sup__fun__def,axiom,
( sup_sup_a_a_o
= ( ^ [F2: a > a > $o,G: a > a > $o,X3: a] : ( sup_sup_a_o @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% sup_fun_def
thf(fact_588_sup__left__commute,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z ) )
= ( sup_sup_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_589_sup__left__commute,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ ( sup_sup_a_a_o @ Y3 @ Z ) )
= ( sup_sup_a_a_o @ Y3 @ ( sup_sup_a_a_o @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_590_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_591_sup_Oleft__commute,axiom,
! [B: a > a > $o,A: a > a > $o,C: a > a > $o] :
( ( sup_sup_a_a_o @ B @ ( sup_sup_a_a_o @ A @ C ) )
= ( sup_sup_a_a_o @ A @ ( sup_sup_a_a_o @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_592_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X3 ) ) ) ).
% sup_commute
thf(fact_593_sup__commute,axiom,
( sup_sup_a_a_o
= ( ^ [X3: a > a > $o,Y: a > a > $o] : ( sup_sup_a_a_o @ Y @ X3 ) ) ) ).
% sup_commute
thf(fact_594_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A2: set_a,B2: set_a] : ( sup_sup_set_a @ B2 @ A2 ) ) ) ).
% sup.commute
thf(fact_595_sup_Ocommute,axiom,
( sup_sup_a_a_o
= ( ^ [A2: a > a > $o,B2: a > a > $o] : ( sup_sup_a_a_o @ B2 @ A2 ) ) ) ).
% sup.commute
thf(fact_596_sup__assoc,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ Z )
= ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z ) ) ) ).
% sup_assoc
thf(fact_597_sup__assoc,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( sup_sup_a_a_o @ ( sup_sup_a_a_o @ X2 @ Y3 ) @ Z )
= ( sup_sup_a_a_o @ X2 @ ( sup_sup_a_a_o @ Y3 @ Z ) ) ) ).
% sup_assoc
thf(fact_598_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_599_sup_Oassoc,axiom,
! [A: a > a > $o,B: a > a > $o,C: a > a > $o] :
( ( sup_sup_a_a_o @ ( sup_sup_a_a_o @ A @ B ) @ C )
= ( sup_sup_a_a_o @ A @ ( sup_sup_a_a_o @ B @ C ) ) ) ).
% sup.assoc
thf(fact_600_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_601_inf__sup__aci_I5_J,axiom,
( sup_sup_a_a_o
= ( ^ [X3: a > a > $o,Y: a > a > $o] : ( sup_sup_a_a_o @ Y @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_602_inf__sup__aci_I6_J,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ Z )
= ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_603_inf__sup__aci_I6_J,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( sup_sup_a_a_o @ ( sup_sup_a_a_o @ X2 @ Y3 ) @ Z )
= ( sup_sup_a_a_o @ X2 @ ( sup_sup_a_a_o @ Y3 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_604_inf__sup__aci_I7_J,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z ) )
= ( sup_sup_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_605_inf__sup__aci_I7_J,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ ( sup_sup_a_a_o @ Y3 @ Z ) )
= ( sup_sup_a_a_o @ Y3 @ ( sup_sup_a_a_o @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_606_inf__sup__aci_I8_J,axiom,
! [X2: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) )
= ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_607_inf__sup__aci_I8_J,axiom,
! [X2: a > a > $o,Y3: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ ( sup_sup_a_a_o @ X2 @ Y3 ) )
= ( sup_sup_a_a_o @ X2 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_608_sup_OcoboundedI2,axiom,
! [C: a > a > $o,B: a > a > $o,A: a > a > $o] :
( ( ord_less_eq_a_a_o @ C @ B )
=> ( ord_less_eq_a_a_o @ C @ ( sup_sup_a_a_o @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_609_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_610_sup_OcoboundedI1,axiom,
! [C: a > a > $o,A: a > a > $o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ C @ A )
=> ( ord_less_eq_a_a_o @ C @ ( sup_sup_a_a_o @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_611_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_612_sup_Oabsorb__iff2,axiom,
( ord_less_eq_a_a_o
= ( ^ [A2: a > a > $o,B2: a > a > $o] :
( ( sup_sup_a_a_o @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_613_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_614_sup_Oabsorb__iff1,axiom,
( ord_less_eq_a_a_o
= ( ^ [B2: a > a > $o,A2: a > a > $o] :
( ( sup_sup_a_a_o @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_615_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_616_sup_Ocobounded2,axiom,
! [B: a > a > $o,A: a > a > $o] : ( ord_less_eq_a_a_o @ B @ ( sup_sup_a_a_o @ A @ B ) ) ).
% sup.cobounded2
thf(fact_617_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_618_sup_Ocobounded1,axiom,
! [A: a > a > $o,B: a > a > $o] : ( ord_less_eq_a_a_o @ A @ ( sup_sup_a_a_o @ A @ B ) ) ).
% sup.cobounded1
thf(fact_619_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_620_sup_Oorder__iff,axiom,
( ord_less_eq_a_a_o
= ( ^ [B2: a > a > $o,A2: a > a > $o] :
( A2
= ( sup_sup_a_a_o @ A2 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_621_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A2: set_a] :
( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_622_sup_OboundedI,axiom,
! [B: a > a > $o,A: a > a > $o,C: a > a > $o] :
( ( ord_less_eq_a_a_o @ B @ A )
=> ( ( ord_less_eq_a_a_o @ C @ A )
=> ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_623_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_624_sup_OboundedE,axiom,
! [B: a > a > $o,C: a > a > $o,A: a > a > $o] :
( ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_a_a_o @ B @ A )
=> ~ ( ord_less_eq_a_a_o @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_625_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_626_sup__absorb2,axiom,
! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ( sup_sup_a_a_o @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_627_sup__absorb2,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( sup_sup_set_a @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_628_sup__absorb1,axiom,
! [Y3: a > a > $o,X2: a > a > $o] :
( ( ord_less_eq_a_a_o @ Y3 @ X2 )
=> ( ( sup_sup_a_a_o @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_629_sup__absorb1,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( sup_sup_set_a @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_630_sup_Oabsorb2,axiom,
! [A: a > a > $o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ A @ B )
=> ( ( sup_sup_a_a_o @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_631_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_632_sup_Oabsorb1,axiom,
! [B: a > a > $o,A: a > a > $o] :
( ( ord_less_eq_a_a_o @ B @ A )
=> ( ( sup_sup_a_a_o @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_633_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_634_sup__unique,axiom,
! [F: ( a > a > $o ) > ( a > a > $o ) > a > a > $o,X2: a > a > $o,Y3: a > a > $o] :
( ! [X: a > a > $o,Y2: a > a > $o] : ( ord_less_eq_a_a_o @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: a > a > $o,Y2: a > a > $o] : ( ord_less_eq_a_a_o @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: a > a > $o,Y2: a > a > $o,Z3: a > a > $o] :
( ( ord_less_eq_a_a_o @ Y2 @ X )
=> ( ( ord_less_eq_a_a_o @ Z3 @ X )
=> ( ord_less_eq_a_a_o @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_a_a_o @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_635_sup__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y3: set_a] :
( ! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ Z3 @ X )
=> ( ord_less_eq_set_a @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_a @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_636_sup_OorderI,axiom,
! [A: a > a > $o,B: a > a > $o] :
( ( A
= ( sup_sup_a_a_o @ A @ B ) )
=> ( ord_less_eq_a_a_o @ B @ A ) ) ).
% sup.orderI
thf(fact_637_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_638_sup_OorderE,axiom,
! [B: a > a > $o,A: a > a > $o] :
( ( ord_less_eq_a_a_o @ B @ A )
=> ( A
= ( sup_sup_a_a_o @ A @ B ) ) ) ).
% sup.orderE
thf(fact_639_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_640_le__iff__sup,axiom,
( ord_less_eq_a_a_o
= ( ^ [X3: a > a > $o,Y: a > a > $o] :
( ( sup_sup_a_a_o @ X3 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_641_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y: set_a] :
( ( sup_sup_set_a @ X3 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_642_sup__least,axiom,
! [Y3: a > a > $o,X2: a > a > $o,Z: a > a > $o] :
( ( ord_less_eq_a_a_o @ Y3 @ X2 )
=> ( ( ord_less_eq_a_a_o @ Z @ X2 )
=> ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ Y3 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_643_sup__least,axiom,
! [Y3: set_a,X2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ Z @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y3 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_644_sup__mono,axiom,
! [A: a > a > $o,C: a > a > $o,B: a > a > $o,D: a > a > $o] :
( ( ord_less_eq_a_a_o @ A @ C )
=> ( ( ord_less_eq_a_a_o @ B @ D )
=> ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ A @ B ) @ ( sup_sup_a_a_o @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_645_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_646_sup_Omono,axiom,
! [C: a > a > $o,A: a > a > $o,D: a > a > $o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ C @ A )
=> ( ( ord_less_eq_a_a_o @ D @ B )
=> ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ C @ D ) @ ( sup_sup_a_a_o @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_647_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_648_le__supI2,axiom,
! [X2: a > a > $o,B: a > a > $o,A: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ B )
=> ( ord_less_eq_a_a_o @ X2 @ ( sup_sup_a_a_o @ A @ B ) ) ) ).
% le_supI2
thf(fact_649_le__supI2,axiom,
! [X2: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X2 @ B )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_650_le__supI1,axiom,
! [X2: a > a > $o,A: a > a > $o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ A )
=> ( ord_less_eq_a_a_o @ X2 @ ( sup_sup_a_a_o @ A @ B ) ) ) ).
% le_supI1
thf(fact_651_le__supI1,axiom,
! [X2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_652_sup__ge2,axiom,
! [Y3: a > a > $o,X2: a > a > $o] : ( ord_less_eq_a_a_o @ Y3 @ ( sup_sup_a_a_o @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_653_sup__ge2,axiom,
! [Y3: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_654_sup__ge1,axiom,
! [X2: a > a > $o,Y3: a > a > $o] : ( ord_less_eq_a_a_o @ X2 @ ( sup_sup_a_a_o @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_655_sup__ge1,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_656_le__supI,axiom,
! [A: a > a > $o,X2: a > a > $o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ A @ X2 )
=> ( ( ord_less_eq_a_a_o @ B @ X2 )
=> ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_657_le__supI,axiom,
! [A: set_a,X2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X2 )
=> ( ( ord_less_eq_set_a @ B @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_658_le__supE,axiom,
! [A: a > a > $o,B: a > a > $o,X2: a > a > $o] :
( ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_a_a_o @ A @ X2 )
=> ~ ( ord_less_eq_a_a_o @ B @ X2 ) ) ) ).
% le_supE
thf(fact_659_le__supE,axiom,
! [A: set_a,B: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_set_a @ A @ X2 )
=> ~ ( ord_less_eq_set_a @ B @ X2 ) ) ) ).
% le_supE
thf(fact_660_inf__sup__ord_I3_J,axiom,
! [X2: a > a > $o,Y3: a > a > $o] : ( ord_less_eq_a_a_o @ X2 @ ( sup_sup_a_a_o @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_661_inf__sup__ord_I3_J,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_662_inf__sup__ord_I4_J,axiom,
! [Y3: a > a > $o,X2: a > a > $o] : ( ord_less_eq_a_a_o @ Y3 @ ( sup_sup_a_a_o @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_663_inf__sup__ord_I4_J,axiom,
! [Y3: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_664_less__supI1,axiom,
! [X2: a > a > $o,A: a > a > $o,B: a > a > $o] :
( ( ord_less_a_a_o @ X2 @ A )
=> ( ord_less_a_a_o @ X2 @ ( sup_sup_a_a_o @ A @ B ) ) ) ).
% less_supI1
thf(fact_665_less__supI1,axiom,
! [X2: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ X2 @ A )
=> ( ord_less_set_a @ X2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI1
thf(fact_666_less__supI2,axiom,
! [X2: a > a > $o,B: a > a > $o,A: a > a > $o] :
( ( ord_less_a_a_o @ X2 @ B )
=> ( ord_less_a_a_o @ X2 @ ( sup_sup_a_a_o @ A @ B ) ) ) ).
% less_supI2
thf(fact_667_less__supI2,axiom,
! [X2: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ X2 @ B )
=> ( ord_less_set_a @ X2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI2
thf(fact_668_sup_Oabsorb3,axiom,
! [B: a > a > $o,A: a > a > $o] :
( ( ord_less_a_a_o @ B @ A )
=> ( ( sup_sup_a_a_o @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_669_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_670_sup_Oabsorb4,axiom,
! [A: a > a > $o,B: a > a > $o] :
( ( ord_less_a_a_o @ A @ B )
=> ( ( sup_sup_a_a_o @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_671_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_672_sup_Ostrict__boundedE,axiom,
! [B: a > a > $o,C: a > a > $o,A: a > a > $o] :
( ( ord_less_a_a_o @ ( sup_sup_a_a_o @ B @ C ) @ A )
=> ~ ( ( ord_less_a_a_o @ B @ A )
=> ~ ( ord_less_a_a_o @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_673_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_674_sup_Ostrict__order__iff,axiom,
( ord_less_a_a_o
= ( ^ [B2: a > a > $o,A2: a > a > $o] :
( ( A2
= ( sup_sup_a_a_o @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_675_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_676_sup_Ostrict__coboundedI1,axiom,
! [C: a > a > $o,A: a > a > $o,B: a > a > $o] :
( ( ord_less_a_a_o @ C @ A )
=> ( ord_less_a_a_o @ C @ ( sup_sup_a_a_o @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_677_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_678_sup_Ostrict__coboundedI2,axiom,
! [C: a > a > $o,B: a > a > $o,A: a > a > $o] :
( ( ord_less_a_a_o @ C @ B )
=> ( ord_less_a_a_o @ C @ ( sup_sup_a_a_o @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_679_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_680_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_681_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_682_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_683_boolean__algebra_Odisj__zero__right,axiom,
! [X2: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ bot_bot_a_a_o )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_684_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_685_boolean__algebra_Odisj__zero__right,axiom,
! [X2: a > $o] :
( ( sup_sup_a_o @ X2 @ bot_bot_a_o )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_686_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_687_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_688_ivl__disj__un__two_I4_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_eq_a @ L @ M )
=> ( ( ord_less_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M ) @ ( set_or5939364468397584554Than_a @ M @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_689_insert__absorb2,axiom,
! [X2: a,A5: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A5 ) )
= ( insert_a @ X2 @ A5 ) ) ).
% insert_absorb2
thf(fact_690_insert__iff,axiom,
! [A: set_a,B: set_a,A5: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A5 ) )
= ( ( A = B )
| ( member_set_a @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_691_insert__iff,axiom,
! [A: a,B: a,A5: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A5 ) )
= ( ( A = B )
| ( member_a @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_692_insertCI,axiom,
! [A: set_a,B4: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B4 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_693_insertCI,axiom,
! [A: a,B4: set_a,B: a] :
( ( ~ ( member_a @ A @ B4 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_694_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_695_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_696_insert__subset,axiom,
! [X2: set_a,A5: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ A5 ) @ B4 )
= ( ( member_set_a @ X2 @ B4 )
& ( ord_le3724670747650509150_set_a @ A5 @ B4 ) ) ) ).
% insert_subset
thf(fact_697_insert__subset,axiom,
! [X2: a,A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A5 ) @ B4 )
= ( ( member_a @ X2 @ B4 )
& ( ord_less_eq_set_a @ A5 @ B4 ) ) ) ).
% insert_subset
thf(fact_698_Un__insert__right,axiom,
! [A5: set_a,A: a,B4: set_a] :
( ( sup_sup_set_a @ A5 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A5 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_699_Un__insert__left,axiom,
! [A: a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_700_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_701_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_702_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_703_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_704_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_705_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_706_singleton__insert__inj__eq_H,axiom,
! [A: a,A5: set_a,B: a] :
( ( ( insert_a @ A @ A5 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A5 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_707_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A5: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A5 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A5 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_708_ivl__subset,axiom,
! [I: a,J: a,M: a,N: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ I @ J ) @ ( set_or5139330845457685135Than_a @ M @ N ) )
= ( ( ord_less_eq_a @ J @ I )
| ( ( ord_less_eq_a @ M @ I )
& ( ord_less_eq_a @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_709_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_710_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_711_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_712_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_713_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_714_atLeastAtMost__singleton,axiom,
! [A: a] :
( ( set_or672772299803893939Most_a @ A @ A )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% atLeastAtMost_singleton
thf(fact_715_mk__disjoint__insert,axiom,
! [A: set_a,A5: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ? [B8: set_set_a] :
( ( A5
= ( insert_set_a @ A @ B8 ) )
& ~ ( member_set_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_716_mk__disjoint__insert,axiom,
! [A: a,A5: set_a] :
( ( member_a @ A @ A5 )
=> ? [B8: set_a] :
( ( A5
= ( insert_a @ A @ B8 ) )
& ~ ( member_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_717_insert__commute,axiom,
! [X2: a,Y3: a,A5: set_a] :
( ( insert_a @ X2 @ ( insert_a @ Y3 @ A5 ) )
= ( insert_a @ Y3 @ ( insert_a @ X2 @ A5 ) ) ) ).
% insert_commute
thf(fact_718_insert__eq__iff,axiom,
! [A: set_a,A5: set_set_a,B: set_a,B4: set_set_a] :
( ~ ( member_set_a @ A @ A5 )
=> ( ~ ( member_set_a @ B @ B4 )
=> ( ( ( insert_set_a @ A @ A5 )
= ( insert_set_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A5 = B4 ) )
& ( ( A != B )
=> ? [C5: set_set_a] :
( ( A5
= ( insert_set_a @ B @ C5 ) )
& ~ ( member_set_a @ B @ C5 )
& ( B4
= ( insert_set_a @ A @ C5 ) )
& ~ ( member_set_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_719_insert__eq__iff,axiom,
! [A: a,A5: set_a,B: a,B4: set_a] :
( ~ ( member_a @ A @ A5 )
=> ( ~ ( member_a @ B @ B4 )
=> ( ( ( insert_a @ A @ A5 )
= ( insert_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A5 = B4 ) )
& ( ( A != B )
=> ? [C5: set_a] :
( ( A5
= ( insert_a @ B @ C5 ) )
& ~ ( member_a @ B @ C5 )
& ( B4
= ( insert_a @ A @ C5 ) )
& ~ ( member_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_720_insert__absorb,axiom,
! [A: set_a,A5: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ( ( insert_set_a @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_721_insert__absorb,axiom,
! [A: a,A5: set_a] :
( ( member_a @ A @ A5 )
=> ( ( insert_a @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_722_insert__ident,axiom,
! [X2: set_a,A5: set_set_a,B4: set_set_a] :
( ~ ( member_set_a @ X2 @ A5 )
=> ( ~ ( member_set_a @ X2 @ B4 )
=> ( ( ( insert_set_a @ X2 @ A5 )
= ( insert_set_a @ X2 @ B4 ) )
= ( A5 = B4 ) ) ) ) ).
% insert_ident
thf(fact_723_insert__ident,axiom,
! [X2: a,A5: set_a,B4: set_a] :
( ~ ( member_a @ X2 @ A5 )
=> ( ~ ( member_a @ X2 @ B4 )
=> ( ( ( insert_a @ X2 @ A5 )
= ( insert_a @ X2 @ B4 ) )
= ( A5 = B4 ) ) ) ) ).
% insert_ident
thf(fact_724_Set_Oset__insert,axiom,
! [X2: set_a,A5: set_set_a] :
( ( member_set_a @ X2 @ A5 )
=> ~ ! [B8: set_set_a] :
( ( A5
= ( insert_set_a @ X2 @ B8 ) )
=> ( member_set_a @ X2 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_725_Set_Oset__insert,axiom,
! [X2: a,A5: set_a] :
( ( member_a @ X2 @ A5 )
=> ~ ! [B8: set_a] :
( ( A5
= ( insert_a @ X2 @ B8 ) )
=> ( member_a @ X2 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_726_insertI2,axiom,
! [A: set_a,B4: set_set_a,B: set_a] :
( ( member_set_a @ A @ B4 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_727_insertI2,axiom,
! [A: a,B4: set_a,B: a] :
( ( member_a @ A @ B4 )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_728_insertI1,axiom,
! [A: set_a,B4: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B4 ) ) ).
% insertI1
thf(fact_729_insertI1,axiom,
! [A: a,B4: set_a] : ( member_a @ A @ ( insert_a @ A @ B4 ) ) ).
% insertI1
thf(fact_730_insertE,axiom,
! [A: set_a,B: set_a,A5: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A5 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A5 ) ) ) ).
% insertE
thf(fact_731_insertE,axiom,
! [A: a,B: a,A5: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A5 ) )
=> ( ( A != B )
=> ( member_a @ A @ A5 ) ) ) ).
% insertE
thf(fact_732_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_733_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_734_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_735_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_736_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_737_insert__not__empty,axiom,
! [A: a,A5: set_a] :
( ( insert_a @ A @ A5 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_738_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_739_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_740_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_741_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_742_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_743_subset__insertI2,axiom,
! [A5: set_a,B4: set_a,B: a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ord_less_eq_set_a @ A5 @ ( insert_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_744_subset__insertI,axiom,
! [B4: set_a,A: a] : ( ord_less_eq_set_a @ B4 @ ( insert_a @ A @ B4 ) ) ).
% subset_insertI
thf(fact_745_subset__insert,axiom,
! [X2: set_a,A5: set_set_a,B4: set_set_a] :
( ~ ( member_set_a @ X2 @ A5 )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ ( insert_set_a @ X2 @ B4 ) )
= ( ord_le3724670747650509150_set_a @ A5 @ B4 ) ) ) ).
% subset_insert
thf(fact_746_subset__insert,axiom,
! [X2: a,A5: set_a,B4: set_a] :
( ~ ( member_a @ X2 @ A5 )
=> ( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X2 @ B4 ) )
= ( ord_less_eq_set_a @ A5 @ B4 ) ) ) ).
% subset_insert
thf(fact_747_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_748_insert__subsetI,axiom,
! [X2: set_a,A5: set_set_a,X6: set_set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( ( ord_le3724670747650509150_set_a @ X6 @ A5 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ X6 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_749_insert__subsetI,axiom,
! [X2: a,A5: set_a,X6: set_a] :
( ( member_a @ X2 @ A5 )
=> ( ( ord_less_eq_set_a @ X6 @ A5 )
=> ( ord_less_eq_set_a @ ( insert_a @ X2 @ X6 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_750_pairwise__insert,axiom,
! [R3: set_a > set_a > $o,X2: set_a,S3: set_set_a] :
( ( pairwise_set_a @ R3 @ ( insert_set_a @ X2 @ S3 ) )
= ( ! [Y: set_a] :
( ( ( member_set_a @ Y @ S3 )
& ( Y != X2 ) )
=> ( ( R3 @ X2 @ Y )
& ( R3 @ Y @ X2 ) ) )
& ( pairwise_set_a @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_751_pairwise__insert,axiom,
! [R3: a > a > $o,X2: a,S3: set_a] :
( ( pairwise_a @ R3 @ ( insert_a @ X2 @ S3 ) )
= ( ! [Y: a] :
( ( ( member_a @ Y @ S3 )
& ( Y != X2 ) )
=> ( ( R3 @ X2 @ Y )
& ( R3 @ Y @ X2 ) ) )
& ( pairwise_a @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_752_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_753_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_754_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_755_ivl__disj__un__two_I3_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_eq_a @ L @ M )
=> ( ( ord_less_eq_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ M ) @ ( set_or5139330845457685135Than_a @ M @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_756_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_757_subset__singletonD,axiom,
! [A5: set_a,X2: a] :
( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ( A5 = bot_bot_set_a )
| ( A5
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_758_singleton__Un__iff,axiom,
! [X2: a,A5: set_a,B4: set_a] :
( ( ( insert_a @ X2 @ bot_bot_set_a )
= ( sup_sup_set_a @ A5 @ B4 ) )
= ( ( ( A5 = bot_bot_set_a )
& ( B4
= ( insert_a @ X2 @ bot_bot_set_a ) ) )
| ( ( A5
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B4 = bot_bot_set_a ) )
| ( ( A5
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B4
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_759_Un__singleton__iff,axiom,
! [A5: set_a,B4: set_a,X2: a] :
( ( ( sup_sup_set_a @ A5 @ B4 )
= ( insert_a @ X2 @ bot_bot_set_a ) )
= ( ( ( A5 = bot_bot_set_a )
& ( B4
= ( insert_a @ X2 @ bot_bot_set_a ) ) )
| ( ( A5
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B4 = bot_bot_set_a ) )
| ( ( A5
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B4
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_760_insert__is__Un,axiom,
( insert_a
= ( ^ [A2: a] : ( sup_sup_set_a @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_761_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_762_pairwise__singleton,axiom,
! [P: a > a > $o,A5: a] : ( pairwise_a @ P @ ( insert_a @ A5 @ bot_bot_set_a ) ) ).
% pairwise_singleton
thf(fact_763_ivl__disj__un__two_I7_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_eq_a @ L @ M )
=> ( ( ord_less_eq_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ M ) @ ( set_or672772299803893939Most_a @ M @ U ) )
= ( set_or672772299803893939Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_764_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_765_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_766_boolean__algebra__cancel_Osup1,axiom,
! [A5: set_a,K: set_a,A: set_a,B: set_a] :
( ( A5
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A5 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_767_boolean__algebra__cancel_Osup1,axiom,
! [A5: a > a > $o,K: a > a > $o,A: a > a > $o,B: a > a > $o] :
( ( A5
= ( sup_sup_a_a_o @ K @ A ) )
=> ( ( sup_sup_a_a_o @ A5 @ B )
= ( sup_sup_a_a_o @ K @ ( sup_sup_a_a_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_768_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_a,K: set_a,B: set_a,A: set_a] :
( ( B4
= ( sup_sup_set_a @ K @ B ) )
=> ( ( sup_sup_set_a @ A @ B4 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_769_boolean__algebra__cancel_Osup2,axiom,
! [B4: a > a > $o,K: a > a > $o,B: a > a > $o,A: a > a > $o] :
( ( B4
= ( sup_sup_a_a_o @ K @ B ) )
=> ( ( sup_sup_a_a_o @ A @ B4 )
= ( sup_sup_a_a_o @ K @ ( sup_sup_a_a_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_770_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_771_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_772_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_773_ivl__disj__un__two__touch_I2_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_eq_a @ L @ M )
=> ( ( ord_less_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M ) @ ( set_or5139330845457685135Than_a @ M @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_774_ivl__disj__un__two_I1_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_a @ L @ M )
=> ( ( ord_less_eq_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M ) @ ( set_or5139330845457685135Than_a @ M @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_775_ivl__disj__un__two__touch_I1_J,axiom,
! [L: a,M: a,U: a] :
( ( ord_less_a @ L @ M )
=> ( ( ord_less_a @ M @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M ) @ ( set_or5139330845457685135Than_a @ M @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_776_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_777_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_778_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_779_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_780_is__singletonE,axiom,
! [A5: set_a] :
( ( is_singleton_a @ A5 )
=> ~ ! [X: a] :
( A5
!= ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_781_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
? [X3: a] :
( A4
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_782_atMost__UNIV__triv,axiom,
( ( set_ord_atMost_set_a @ top_top_set_a )
= top_top_set_set_a ) ).
% atMost_UNIV_triv
thf(fact_783_UNIV__I,axiom,
! [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_784_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_785_atLeast__empty__triv,axiom,
( ( set_or8362275514725411625_set_a @ bot_bot_set_a )
= top_top_set_set_a ) ).
% atLeast_empty_triv
thf(fact_786_sup__top__left,axiom,
! [X2: a > a > $o] :
( ( sup_sup_a_a_o @ top_top_a_a_o @ X2 )
= top_top_a_a_o ) ).
% sup_top_left
thf(fact_787_sup__top__left,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X2 )
= top_top_set_a ) ).
% sup_top_left
thf(fact_788_sup__top__right,axiom,
! [X2: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ top_top_a_a_o )
= top_top_a_a_o ) ).
% sup_top_right
thf(fact_789_sup__top__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ top_top_set_a )
= top_top_set_a ) ).
% sup_top_right
thf(fact_790_boolean__algebra_Odisj__one__right,axiom,
! [X2: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ top_top_a_a_o )
= top_top_a_a_o ) ).
% boolean_algebra.disj_one_right
thf(fact_791_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ top_top_set_a )
= top_top_set_a ) ).
% boolean_algebra.disj_one_right
thf(fact_792_boolean__algebra_Odisj__one__left,axiom,
! [X2: a > a > $o] :
( ( sup_sup_a_a_o @ top_top_a_a_o @ X2 )
= top_top_a_a_o ) ).
% boolean_algebra.disj_one_left
thf(fact_793_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X2 )
= top_top_set_a ) ).
% boolean_algebra.disj_one_left
thf(fact_794_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_795_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_796_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_797_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_798_top_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ top_top_set_a @ A ) ).
% top.extremum_strict
thf(fact_799_UNIV__eq__I,axiom,
! [A5: set_set_a] :
( ! [X: set_a] : ( member_set_a @ X @ A5 )
=> ( top_top_set_set_a = A5 ) ) ).
% UNIV_eq_I
thf(fact_800_UNIV__eq__I,axiom,
! [A5: set_a] :
( ! [X: a] : ( member_a @ X @ A5 )
=> ( top_top_set_a = A5 ) ) ).
% UNIV_eq_I
thf(fact_801_UNIV__witness,axiom,
? [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_802_UNIV__witness,axiom,
? [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_witness
thf(fact_803_atMost__eq__UNIV__iff,axiom,
! [X2: set_a] :
( ( ( set_ord_atMost_set_a @ X2 )
= top_top_set_set_a )
= ( X2 = top_top_set_a ) ) ).
% atMost_eq_UNIV_iff
thf(fact_804_atLeastAtMost__eq__UNIV__iff,axiom,
! [X2: a > $o,Y3: a > $o] :
( ( ( set_or497184483940929162st_a_o @ X2 @ Y3 )
= top_top_set_a_o )
= ( ( X2 = bot_bot_a_o )
& ( Y3 = top_top_a_o ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_805_atLeastAtMost__eq__UNIV__iff,axiom,
! [X2: set_a,Y3: set_a] :
( ( ( set_or6288561110385358355_set_a @ X2 @ Y3 )
= top_top_set_set_a )
= ( ( X2 = bot_bot_set_a )
& ( Y3 = top_top_set_a ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_806_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_807_subset__UNIV,axiom,
! [A5: set_a] : ( ord_less_eq_set_a @ A5 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_808_insert__UNIV,axiom,
! [X2: a] :
( ( insert_a @ X2 @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_809_Un__UNIV__left,axiom,
! [B4: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ B4 )
= top_top_set_a ) ).
% Un_UNIV_left
thf(fact_810_Un__UNIV__right,axiom,
! [A5: set_a] :
( ( sup_sup_set_a @ A5 @ top_top_set_a )
= top_top_set_a ) ).
% Un_UNIV_right
thf(fact_811_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
( A4
= ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_812_atLeast__eq__UNIV__iff,axiom,
! [X2: set_a] :
( ( ( set_or8362275514725411625_set_a @ X2 )
= top_top_set_set_a )
= ( X2 = bot_bot_set_a ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_813_atLeast__eq__UNIV__iff,axiom,
! [X2: a > $o] :
( ( ( set_ord_atLeast_a_o @ X2 )
= top_top_set_a_o )
= ( X2 = bot_bot_a_o ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_814_is__singletonI_H,axiom,
! [A5: set_set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ( member_set_a @ Y2 @ A5 )
=> ( X = Y2 ) ) )
=> ( is_singleton_set_a @ A5 ) ) ) ).
% is_singletonI'
thf(fact_815_is__singletonI_H,axiom,
! [A5: set_a] :
( ( A5 != bot_bot_set_a )
=> ( ! [X: a,Y2: a] :
( ( member_a @ X @ A5 )
=> ( ( member_a @ Y2 @ A5 )
=> ( X = Y2 ) ) )
=> ( is_singleton_a @ A5 ) ) ) ).
% is_singletonI'
thf(fact_816_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_817_Inf__atMostLessThan,axiom,
! [X2: a > $o] :
( ( ord_less_a_o @ top_top_a_o @ X2 )
=> ( ( complete_Inf_Inf_a_o @ ( set_ord_lessThan_a_o @ X2 ) )
= bot_bot_a_o ) ) ).
% Inf_atMostLessThan
thf(fact_818_Inf__atMostLessThan,axiom,
! [X2: set_a] :
( ( ord_less_set_a @ top_top_set_a @ X2 )
=> ( ( comple6135023378680113637_set_a @ ( set_or5421148953861284865_set_a @ X2 ) )
= bot_bot_set_a ) ) ).
% Inf_atMostLessThan
thf(fact_819_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_820_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_821_Int__iff,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B4 ) )
= ( ( member_set_a @ C @ A5 )
& ( member_set_a @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_822_Int__iff,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B4 ) )
= ( ( member_a @ C @ A5 )
& ( member_a @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_823_IntI,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ A5 )
=> ( ( member_set_a @ C @ B4 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B4 ) ) ) ) ).
% IntI
thf(fact_824_IntI,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ A5 )
=> ( ( member_a @ C @ B4 )
=> ( member_a @ C @ ( inf_inf_set_a @ A5 @ B4 ) ) ) ) ).
% IntI
thf(fact_825_Diff__idemp,axiom,
! [A5: set_a,B4: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A5 @ B4 ) @ B4 )
= ( minus_minus_set_a @ A5 @ B4 ) ) ).
% Diff_idemp
thf(fact_826_Diff__iff,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B4 ) )
= ( ( member_set_a @ C @ A5 )
& ~ ( member_set_a @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_827_Diff__iff,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B4 ) )
= ( ( member_a @ C @ A5 )
& ~ ( member_a @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_828_DiffI,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ A5 )
=> ( ~ ( member_set_a @ C @ B4 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B4 ) ) ) ) ).
% DiffI
thf(fact_829_DiffI,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ A5 )
=> ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ ( minus_minus_set_a @ A5 @ B4 ) ) ) ) ).
% DiffI
thf(fact_830_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_831_le__inf__iff,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_832_inf__bot__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_833_inf__bot__left,axiom,
! [X2: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X2 )
= bot_bot_a_o ) ).
% inf_bot_left
thf(fact_834_inf__bot__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_835_inf__bot__right,axiom,
! [X2: a > $o] :
( ( inf_inf_a_o @ X2 @ bot_bot_a_o )
= bot_bot_a_o ) ).
% inf_bot_right
thf(fact_836_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_837_boolean__algebra_Oconj__zero__left,axiom,
! [X2: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X2 )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_838_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_839_boolean__algebra_Oconj__zero__right,axiom,
! [X2: a > $o] :
( ( inf_inf_a_o @ X2 @ bot_bot_a_o )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_840_sup__inf__absorb,axiom,
! [X2: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y3 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_841_sup__inf__absorb,axiom,
! [X2: a > a > $o,Y3: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ ( inf_inf_a_a_o @ X2 @ Y3 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_842_inf__sup__absorb,axiom,
! [X2: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_843_inf__sup__absorb,axiom,
! [X2: a > a > $o,Y3: a > a > $o] :
( ( inf_inf_a_a_o @ X2 @ ( sup_sup_a_a_o @ X2 @ Y3 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_844_Int__UNIV,axiom,
! [A5: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A5 @ B4 )
= top_top_set_a )
= ( ( A5 = top_top_set_a )
& ( B4 = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_845_Diff__cancel,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ A5 @ A5 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_846_empty__Diff,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A5 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_847_Diff__empty,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ A5 @ bot_bot_set_a )
= A5 ) ).
% Diff_empty
thf(fact_848_Int__subset__iff,axiom,
! [C2: set_a,A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A5 @ B4 ) )
= ( ( ord_less_eq_set_a @ C2 @ A5 )
& ( ord_less_eq_set_a @ C2 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_849_Int__insert__right__if1,axiom,
! [A: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B4 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A5 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_850_Int__insert__right__if1,axiom,
! [A: a,A5: set_a,B4: set_a] :
( ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A5 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_851_Int__insert__right__if0,axiom,
! [A: set_a,A5: set_set_a,B4: set_set_a] :
( ~ ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B4 ) )
= ( inf_inf_set_set_a @ A5 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_852_Int__insert__right__if0,axiom,
! [A: a,A5: set_a,B4: set_a] :
( ~ ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B4 ) )
= ( inf_inf_set_a @ A5 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_853_insert__inter__insert,axiom,
! [A: a,A5: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A5 ) @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A5 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_854_Int__insert__left__if1,axiom,
! [A: set_a,C2: set_set_a,B4: set_set_a] :
( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B4 ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_855_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B4: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_856_Int__insert__left__if0,axiom,
! [A: set_a,C2: set_set_a,B4: set_set_a] :
( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_set_a @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_857_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B4: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_858_insert__Diff1,axiom,
! [X2: set_a,B4: set_set_a,A5: set_set_a] :
( ( member_set_a @ X2 @ B4 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A5 ) @ B4 )
= ( minus_5736297505244876581_set_a @ A5 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_859_insert__Diff1,axiom,
! [X2: a,B4: set_a,A5: set_a] :
( ( member_a @ X2 @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A5 ) @ B4 )
= ( minus_minus_set_a @ A5 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_860_Diff__insert0,axiom,
! [X2: set_a,A5: set_set_a,B4: set_set_a] :
( ~ ( member_set_a @ X2 @ A5 )
=> ( ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X2 @ B4 ) )
= ( minus_5736297505244876581_set_a @ A5 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_861_Diff__insert0,axiom,
! [X2: a,A5: set_a,B4: set_a] :
( ~ ( member_a @ X2 @ A5 )
=> ( ( minus_minus_set_a @ A5 @ ( insert_a @ X2 @ B4 ) )
= ( minus_minus_set_a @ A5 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_862_Un__Int__eq_I1_J,axiom,
! [S2: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_863_Un__Int__eq_I2_J,axiom,
! [S2: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_864_Un__Int__eq_I3_J,axiom,
! [S2: set_a,T2: set_a] :
( ( inf_inf_set_a @ S2 @ ( sup_sup_set_a @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_865_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S2: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_866_Int__Un__eq_I1_J,axiom,
! [S2: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_867_Int__Un__eq_I2_J,axiom,
! [S2: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_868_Int__Un__eq_I3_J,axiom,
! [S2: set_a,T2: set_a] :
( ( sup_sup_set_a @ S2 @ ( inf_inf_set_a @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_869_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S2: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_870_Un__Diff__cancel2,axiom,
! [B4: set_a,A5: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B4 @ A5 ) @ A5 )
= ( sup_sup_set_a @ B4 @ A5 ) ) ).
% Un_Diff_cancel2
thf(fact_871_Un__Diff__cancel,axiom,
! [A5: set_a,B4: set_a] :
( ( sup_sup_set_a @ A5 @ ( minus_minus_set_a @ B4 @ A5 ) )
= ( sup_sup_set_a @ A5 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_872_Diff__UNIV,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ A5 @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_873_cInf__atLeastAtMost,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ Y3 @ X2 ) )
= Y3 ) ) ).
% cInf_atLeastAtMost
thf(fact_874_Inf__atLeastAtMost,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ X2 @ Y3 ) )
= X2 ) ) ).
% Inf_atLeastAtMost
thf(fact_875_insert__disjoint_I1_J,axiom,
! [A: set_a,A5: set_set_a,B4: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A5 ) @ B4 )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B4 )
& ( ( inf_inf_set_set_a @ A5 @ B4 )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_876_insert__disjoint_I1_J,axiom,
! [A: a,A5: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A5 ) @ B4 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B4 )
& ( ( inf_inf_set_a @ A5 @ B4 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_877_insert__disjoint_I2_J,axiom,
! [A: set_a,A5: set_set_a,B4: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A5 ) @ B4 ) )
= ( ~ ( member_set_a @ A @ B4 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A5 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_878_insert__disjoint_I2_J,axiom,
! [A: a,A5: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A5 ) @ B4 ) )
= ( ~ ( member_a @ A @ B4 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_879_disjoint__insert_I1_J,axiom,
! [B4: set_set_a,A: set_a,A5: set_set_a] :
( ( ( inf_inf_set_set_a @ B4 @ ( insert_set_a @ A @ A5 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B4 )
& ( ( inf_inf_set_set_a @ B4 @ A5 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_880_disjoint__insert_I1_J,axiom,
! [B4: set_a,A: a,A5: set_a] :
( ( ( inf_inf_set_a @ B4 @ ( insert_a @ A @ A5 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B4 )
& ( ( inf_inf_set_a @ B4 @ A5 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_881_disjoint__insert_I2_J,axiom,
! [A5: set_set_a,B: set_a,B4: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ B @ B4 ) ) )
= ( ~ ( member_set_a @ B @ A5 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A5 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_882_disjoint__insert_I2_J,axiom,
! [A5: set_a,B: a,B4: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ ( insert_a @ B @ B4 ) ) )
= ( ~ ( member_a @ B @ A5 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_883_Diff__eq__empty__iff,axiom,
! [A5: set_a,B4: set_a] :
( ( ( minus_minus_set_a @ A5 @ B4 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A5 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_884_insert__Diff__single,axiom,
! [A: a,A5: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A5 ) ) ).
% insert_Diff_single
thf(fact_885_ivl__diff,axiom,
! [I: a,N: a,M: a] :
( ( ord_less_eq_a @ I @ N )
=> ( ( minus_minus_set_a @ ( set_or5139330845457685135Than_a @ I @ M ) @ ( set_or5139330845457685135Than_a @ I @ N ) )
= ( set_or5139330845457685135Than_a @ N @ M ) ) ) ).
% ivl_diff
thf(fact_886_cInf__atLeastLessThan,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_set_a @ Y3 @ X2 )
=> ( ( comple6135023378680113637_set_a @ ( set_or2348907005316661231_set_a @ Y3 @ X2 ) )
= Y3 ) ) ).
% cInf_atLeastLessThan
thf(fact_887_Inf__atLeastLessThan,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ( comple6135023378680113637_set_a @ ( set_or2348907005316661231_set_a @ X2 @ Y3 ) )
= X2 ) ) ).
% Inf_atLeastLessThan
thf(fact_888_Diff__disjoint,axiom,
! [A5: set_a,B4: set_a] :
( ( inf_inf_set_a @ A5 @ ( minus_minus_set_a @ B4 @ A5 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_889_Inf__atMost,axiom,
! [X2: set_a] :
( ( comple6135023378680113637_set_a @ ( set_ord_atMost_set_a @ X2 ) )
= bot_bot_set_a ) ).
% Inf_atMost
thf(fact_890_Inf__atMost,axiom,
! [X2: a > $o] :
( ( complete_Inf_Inf_a_o @ ( set_ord_atMost_a_o @ X2 ) )
= bot_bot_a_o ) ).
% Inf_atMost
thf(fact_891_lessThan__minus__lessThan,axiom,
! [N: a,M: a] :
( ( minus_minus_set_a @ ( set_ord_lessThan_a @ N ) @ ( set_ord_lessThan_a @ M ) )
= ( set_or5139330845457685135Than_a @ M @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_892_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_893_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_894_Int__UNIV__left,axiom,
! [B4: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_895_Int__UNIV__right,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ top_top_set_a )
= A5 ) ).
% Int_UNIV_right
thf(fact_896_Int__mono,axiom,
! [A5: set_a,C2: set_a,B4: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A5 @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B4 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_897_Int__lower1,axiom,
! [A5: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B4 ) @ A5 ) ).
% Int_lower1
thf(fact_898_Int__lower2,axiom,
! [A5: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_899_Int__absorb1,axiom,
! [B4: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B4 @ A5 )
=> ( ( inf_inf_set_a @ A5 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_900_Int__absorb2,axiom,
! [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( inf_inf_set_a @ A5 @ B4 )
= A5 ) ) ).
% Int_absorb2
thf(fact_901_Int__greatest,axiom,
! [C2: set_a,A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ A5 )
=> ( ( ord_less_eq_set_a @ C2 @ B4 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A5 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_902_Int__Collect__mono,axiom,
! [A5: set_set_a,B4: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A5 @ B4 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A5 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B4 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_903_Int__Collect__mono,axiom,
! [A5: set_a,B4: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B4 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_904_Int__insert__left,axiom,
! [A: set_a,C2: set_set_a,B4: set_set_a] :
( ( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B4 ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B4 @ C2 ) ) ) )
& ( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_set_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_905_Int__insert__left,axiom,
! [A: a,C2: set_a,B4: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_906_Int__insert__right,axiom,
! [A: set_a,A5: set_set_a,B4: set_set_a] :
( ( ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B4 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A5 @ B4 ) ) ) )
& ( ~ ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B4 ) )
= ( inf_inf_set_set_a @ A5 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_907_Int__insert__right,axiom,
! [A: a,A5: set_a,B4: set_a] :
( ( ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A5 @ B4 ) ) ) )
& ( ~ ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B4 ) )
= ( inf_inf_set_a @ A5 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_908_Diff__mono,axiom,
! [A5: set_a,C2: set_a,D2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B4 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ B4 ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_909_Diff__subset,axiom,
! [A5: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ B4 ) @ A5 ) ).
% Diff_subset
thf(fact_910_double__diff,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ( minus_minus_set_a @ B4 @ ( minus_minus_set_a @ C2 @ A5 ) )
= A5 ) ) ) ).
% double_diff
thf(fact_911_insert__Diff__if,axiom,
! [X2: set_a,B4: set_set_a,A5: set_set_a] :
( ( ( member_set_a @ X2 @ B4 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A5 ) @ B4 )
= ( minus_5736297505244876581_set_a @ A5 @ B4 ) ) )
& ( ~ ( member_set_a @ X2 @ B4 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A5 ) @ B4 )
= ( insert_set_a @ X2 @ ( minus_5736297505244876581_set_a @ A5 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_912_insert__Diff__if,axiom,
! [X2: a,B4: set_a,A5: set_a] :
( ( ( member_a @ X2 @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A5 ) @ B4 )
= ( minus_minus_set_a @ A5 @ B4 ) ) )
& ( ~ ( member_a @ X2 @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A5 ) @ B4 )
= ( insert_a @ X2 @ ( minus_minus_set_a @ A5 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_913_Un__Int__crazy,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B4 ) @ ( inf_inf_set_a @ B4 @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A5 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A5 @ B4 ) @ ( sup_sup_set_a @ B4 @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A5 ) ) ) ).
% Un_Int_crazy
thf(fact_914_Int__Un__distrib,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ A5 @ ( sup_sup_set_a @ B4 @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B4 ) @ ( inf_inf_set_a @ A5 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_915_Un__Int__distrib,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ A5 @ ( inf_inf_set_a @ B4 @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A5 @ B4 ) @ ( sup_sup_set_a @ A5 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_916_Int__Un__distrib2,axiom,
! [B4: set_a,C2: set_a,A5: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B4 @ C2 ) @ A5 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B4 @ A5 ) @ ( inf_inf_set_a @ C2 @ A5 ) ) ) ).
% Int_Un_distrib2
thf(fact_917_Un__Int__distrib2,axiom,
! [B4: set_a,C2: set_a,A5: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B4 @ C2 ) @ A5 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B4 @ A5 ) @ ( sup_sup_set_a @ C2 @ A5 ) ) ) ).
% Un_Int_distrib2
thf(fact_918_Un__Diff,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A5 @ B4 ) @ C2 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A5 @ C2 ) @ ( minus_minus_set_a @ B4 @ C2 ) ) ) ).
% Un_Diff
thf(fact_919_psubset__imp__ex__mem,axiom,
! [A5: set_set_a,B4: set_set_a] :
( ( ord_less_set_set_a @ A5 @ B4 )
=> ? [B3: set_a] : ( member_set_a @ B3 @ ( minus_5736297505244876581_set_a @ B4 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_920_psubset__imp__ex__mem,axiom,
! [A5: set_a,B4: set_a] :
( ( ord_less_set_a @ A5 @ B4 )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B4 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_921_Diff__Int__distrib2,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A5 @ B4 ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ C2 ) @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_922_Int__left__commute,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ B4 @ C2 ) )
= ( inf_inf_set_a @ B4 @ ( inf_inf_set_a @ A5 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_923_Diff__Int__distrib,axiom,
! [C2: set_a,A5: set_a,B4: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A5 @ B4 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A5 ) @ ( inf_inf_set_a @ C2 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_924_Int__left__absorb,axiom,
! [A5: set_a,B4: set_a] :
( ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ A5 @ B4 ) )
= ( inf_inf_set_a @ A5 @ B4 ) ) ).
% Int_left_absorb
thf(fact_925_Diff__Diff__Int,axiom,
! [A5: set_a,B4: set_a] :
( ( minus_minus_set_a @ A5 @ ( minus_minus_set_a @ A5 @ B4 ) )
= ( inf_inf_set_a @ A5 @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_926_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A4 ) ) ) ).
% Int_commute
thf(fact_927_Int__absorb,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ A5 )
= A5 ) ).
% Int_absorb
thf(fact_928_Int__assoc,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B4 ) @ C2 )
= ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ).
% Int_assoc
thf(fact_929_Diff__Int2,axiom,
! [A5: set_a,C2: set_a,B4: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ C2 ) @ ( inf_inf_set_a @ B4 @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ C2 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_930_Int__Diff,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ B4 ) @ C2 )
= ( inf_inf_set_a @ A5 @ ( minus_minus_set_a @ B4 @ C2 ) ) ) ).
% Int_Diff
thf(fact_931_DiffD2,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B4 ) )
=> ~ ( member_set_a @ C @ B4 ) ) ).
% DiffD2
thf(fact_932_DiffD2,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B4 ) )
=> ~ ( member_a @ C @ B4 ) ) ).
% DiffD2
thf(fact_933_DiffD1,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B4 ) )
=> ( member_set_a @ C @ A5 ) ) ).
% DiffD1
thf(fact_934_DiffD1,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B4 ) )
=> ( member_a @ C @ A5 ) ) ).
% DiffD1
thf(fact_935_IntD2,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B4 ) )
=> ( member_set_a @ C @ B4 ) ) ).
% IntD2
thf(fact_936_IntD2,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B4 ) )
=> ( member_a @ C @ B4 ) ) ).
% IntD2
thf(fact_937_IntD1,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B4 ) )
=> ( member_set_a @ C @ A5 ) ) ).
% IntD1
thf(fact_938_IntD1,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B4 ) )
=> ( member_a @ C @ A5 ) ) ).
% IntD1
thf(fact_939_DiffE,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B4 ) )
=> ~ ( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B4 ) ) ) ).
% DiffE
thf(fact_940_DiffE,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B4 ) )
=> ~ ( ( member_a @ C @ A5 )
=> ( member_a @ C @ B4 ) ) ) ).
% DiffE
thf(fact_941_IntE,axiom,
! [C: set_a,A5: set_set_a,B4: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B4 ) )
=> ~ ( ( member_set_a @ C @ A5 )
=> ~ ( member_set_a @ C @ B4 ) ) ) ).
% IntE
thf(fact_942_IntE,axiom,
! [C: a,A5: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B4 ) )
=> ~ ( ( member_a @ C @ A5 )
=> ~ ( member_a @ C @ B4 ) ) ) ).
% IntE
thf(fact_943_Diff__triv,axiom,
! [A5: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A5 @ B4 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A5 @ B4 )
= A5 ) ) ).
% Diff_triv
thf(fact_944_Int__Diff__disjoint,axiom,
! [A5: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B4 ) @ ( minus_minus_set_a @ A5 @ B4 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_945_Int__emptyI,axiom,
! [A5: set_set_a,B4: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ~ ( member_set_a @ X @ B4 ) )
=> ( ( inf_inf_set_set_a @ A5 @ B4 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_946_Int__emptyI,axiom,
! [A5: set_a,B4: set_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ~ ( member_a @ X @ B4 ) )
=> ( ( inf_inf_set_a @ A5 @ B4 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_947_disjoint__iff,axiom,
! [A5: set_set_a,B4: set_set_a] :
( ( ( inf_inf_set_set_a @ A5 @ B4 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ~ ( member_set_a @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_948_disjoint__iff,axiom,
! [A5: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A5 @ B4 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ~ ( member_a @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_949_Int__empty__left,axiom,
! [B4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B4 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_950_Int__empty__right,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_951_disjoint__iff__not__equal,axiom,
! [A5: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A5 @ B4 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ! [Y: a] :
( ( member_a @ Y @ B4 )
=> ( X3 != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_952_distrib__imp1,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ! [X: set_a,Y2: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y2 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ ( inf_inf_set_a @ X @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_953_distrib__imp1,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ! [X: a > a > $o,Y2: a > a > $o,Z3: a > a > $o] :
( ( inf_inf_a_a_o @ X @ ( sup_sup_a_a_o @ Y2 @ Z3 ) )
= ( sup_sup_a_a_o @ ( inf_inf_a_a_o @ X @ Y2 ) @ ( inf_inf_a_a_o @ X @ Z3 ) ) )
=> ( ( sup_sup_a_a_o @ X2 @ ( inf_inf_a_a_o @ Y3 @ Z ) )
= ( inf_inf_a_a_o @ ( sup_sup_a_a_o @ X2 @ Y3 ) @ ( sup_sup_a_a_o @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_954_distrib__imp2,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ! [X: set_a,Y2: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y2 ) @ ( sup_sup_set_a @ X @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_955_distrib__imp2,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ! [X: a > a > $o,Y2: a > a > $o,Z3: a > a > $o] :
( ( sup_sup_a_a_o @ X @ ( inf_inf_a_a_o @ Y2 @ Z3 ) )
= ( inf_inf_a_a_o @ ( sup_sup_a_a_o @ X @ Y2 ) @ ( sup_sup_a_a_o @ X @ Z3 ) ) )
=> ( ( inf_inf_a_a_o @ X2 @ ( sup_sup_a_a_o @ Y3 @ Z ) )
= ( sup_sup_a_a_o @ ( inf_inf_a_a_o @ X2 @ Y3 ) @ ( inf_inf_a_a_o @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_956_inf__sup__distrib1,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_957_inf__sup__distrib1,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( inf_inf_a_a_o @ X2 @ ( sup_sup_a_a_o @ Y3 @ Z ) )
= ( sup_sup_a_a_o @ ( inf_inf_a_a_o @ X2 @ Y3 ) @ ( inf_inf_a_a_o @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_958_inf__sup__distrib2,axiom,
! [Y3: set_a,Z: set_a,X2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ Z ) @ X2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ X2 ) @ ( inf_inf_set_a @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_959_inf__sup__distrib2,axiom,
! [Y3: a > a > $o,Z: a > a > $o,X2: a > a > $o] :
( ( inf_inf_a_a_o @ ( sup_sup_a_a_o @ Y3 @ Z ) @ X2 )
= ( sup_sup_a_a_o @ ( inf_inf_a_a_o @ Y3 @ X2 ) @ ( inf_inf_a_a_o @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_960_sup__inf__distrib1,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_961_sup__inf__distrib1,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ ( inf_inf_a_a_o @ Y3 @ Z ) )
= ( inf_inf_a_a_o @ ( sup_sup_a_a_o @ X2 @ Y3 ) @ ( sup_sup_a_a_o @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_962_sup__inf__distrib2,axiom,
! [Y3: set_a,Z: set_a,X2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ Z ) @ X2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ X2 ) @ ( sup_sup_set_a @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_963_sup__inf__distrib2,axiom,
! [Y3: a > a > $o,Z: a > a > $o,X2: a > a > $o] :
( ( sup_sup_a_a_o @ ( inf_inf_a_a_o @ Y3 @ Z ) @ X2 )
= ( inf_inf_a_a_o @ ( sup_sup_a_a_o @ Y3 @ X2 ) @ ( sup_sup_a_a_o @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_964_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_965_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_966_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_967_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_968_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_969_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_970_less__infI2,axiom,
! [B: set_a,X2: set_a,A: set_a] :
( ( ord_less_set_a @ B @ X2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ X2 ) ) ).
% less_infI2
thf(fact_971_less__infI1,axiom,
! [A: set_a,X2: set_a,B: set_a] :
( ( ord_less_set_a @ A @ X2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ X2 ) ) ).
% less_infI1
thf(fact_972_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_973_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_974_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_975_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_976_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_977_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_978_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_979_inf__greatest,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ X2 @ Z )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_980_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_981_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_982_inf__absorb2,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( inf_inf_set_a @ X2 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_983_inf__absorb1,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( inf_inf_set_a @ X2 @ Y3 )
= X2 ) ) ).
% inf_absorb1
thf(fact_984_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_985_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_986_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y: set_a] :
( ( inf_inf_set_a @ X3 @ Y )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_987_inf__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y3: set_a] :
( ! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y2 ) @ X )
=> ( ! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y2 ) @ Y2 )
=> ( ! [X: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ X @ Z3 )
=> ( ord_less_eq_set_a @ X @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_988_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_989_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_990_le__infI2,axiom,
! [B: set_a,X2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_991_le__infI1,axiom,
! [A: set_a,X2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_992_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_993_le__infI,axiom,
! [X2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ B )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_994_le__infE,axiom,
! [X2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X2 @ A )
=> ~ ( ord_less_eq_set_a @ X2 @ B ) ) ) ).
% le_infE
thf(fact_995_inf__le2,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_996_inf__le1,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_997_inf__sup__ord_I1_J,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_998_inf__sup__ord_I2_J,axiom,
! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_999_cInf__eq__minimum,axiom,
! [Z: set_a,X6: set_set_a] :
( ( member_set_a @ Z @ X6 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ X6 )
=> ( ord_less_eq_set_a @ Z @ X ) )
=> ( ( comple6135023378680113637_set_a @ X6 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1000_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: set_a,Z: set_a,X2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ Z ) @ X2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ X2 ) @ ( sup_sup_set_a @ Z @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1001_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: a > a > $o,Z: a > a > $o,X2: a > a > $o] :
( ( sup_sup_a_a_o @ ( inf_inf_a_a_o @ Y3 @ Z ) @ X2 )
= ( inf_inf_a_a_o @ ( sup_sup_a_a_o @ Y3 @ X2 ) @ ( sup_sup_a_a_o @ Z @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1002_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: set_a,Z: set_a,X2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ Z ) @ X2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ X2 ) @ ( inf_inf_set_a @ Z @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1003_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: a > a > $o,Z: a > a > $o,X2: a > a > $o] :
( ( inf_inf_a_a_o @ ( sup_sup_a_a_o @ Y3 @ Z ) @ X2 )
= ( sup_sup_a_a_o @ ( inf_inf_a_a_o @ Y3 @ X2 ) @ ( inf_inf_a_a_o @ Z @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1004_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1005_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( sup_sup_a_a_o @ X2 @ ( inf_inf_a_a_o @ Y3 @ Z ) )
= ( inf_inf_a_a_o @ ( sup_sup_a_a_o @ X2 @ Y3 ) @ ( sup_sup_a_a_o @ X2 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1006_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1007_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( inf_inf_a_a_o @ X2 @ ( sup_sup_a_a_o @ Y3 @ Z ) )
= ( sup_sup_a_a_o @ ( inf_inf_a_a_o @ X2 @ Y3 ) @ ( inf_inf_a_a_o @ X2 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1008_Diff__Un,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( minus_minus_set_a @ A5 @ ( sup_sup_set_a @ B4 @ C2 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A5 @ B4 ) @ ( minus_minus_set_a @ A5 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1009_Diff__Int,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( minus_minus_set_a @ A5 @ ( inf_inf_set_a @ B4 @ C2 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A5 @ B4 ) @ ( minus_minus_set_a @ A5 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1010_Int__Diff__Un,axiom,
! [A5: set_a,B4: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B4 ) @ ( minus_minus_set_a @ A5 @ B4 ) )
= A5 ) ).
% Int_Diff_Un
thf(fact_1011_Un__Diff__Int,axiom,
! [A5: set_a,B4: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A5 @ B4 ) @ ( inf_inf_set_a @ A5 @ B4 ) )
= A5 ) ).
% Un_Diff_Int
thf(fact_1012_cInf__greatest,axiom,
! [X6: set_set_a,Z: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ X6 )
=> ( ord_less_eq_set_a @ Z @ X ) )
=> ( ord_less_eq_set_a @ Z @ ( comple6135023378680113637_set_a @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_1013_cInf__eq__non__empty,axiom,
! [X6: set_set_a,A: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ X6 )
=> ( ord_less_eq_set_a @ A @ X ) )
=> ( ! [Y2: set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ord_less_eq_set_a @ Y2 @ X4 ) )
=> ( ord_less_eq_set_a @ Y2 @ A ) )
=> ( ( comple6135023378680113637_set_a @ X6 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1014_distrib__inf__le,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] : ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ ( inf_inf_a_a_o @ X2 @ Y3 ) @ ( inf_inf_a_a_o @ X2 @ Z ) ) @ ( inf_inf_a_a_o @ X2 @ ( sup_sup_a_a_o @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1015_distrib__inf__le,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z ) ) @ ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1016_distrib__sup__le,axiom,
! [X2: a > a > $o,Y3: a > a > $o,Z: a > a > $o] : ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ X2 @ ( inf_inf_a_a_o @ Y3 @ Z ) ) @ ( inf_inf_a_a_o @ ( sup_sup_a_a_o @ X2 @ Y3 ) @ ( sup_sup_a_a_o @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1017_distrib__sup__le,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1018_diff__shunt__var,axiom,
! [X2: a > $o,Y3: a > $o] :
( ( ( minus_minus_a_o @ X2 @ Y3 )
= bot_bot_a_o )
= ( ord_less_eq_a_o @ X2 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1019_diff__shunt__var,axiom,
! [X2: set_a,Y3: set_a] :
( ( ( minus_minus_set_a @ X2 @ Y3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1020_Diff__insert,axiom,
! [A5: set_a,A: a,B4: set_a] :
( ( minus_minus_set_a @ A5 @ ( insert_a @ A @ B4 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A5 @ B4 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_1021_insert__Diff,axiom,
! [A: set_a,A5: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_1022_insert__Diff,axiom,
! [A: a,A5: set_a] :
( ( member_a @ A @ A5 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_1023_Diff__insert2,axiom,
! [A5: set_a,A: a,B4: set_a] :
( ( minus_minus_set_a @ A5 @ ( insert_a @ A @ B4 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_1024_Diff__insert__absorb,axiom,
! [X2: set_a,A5: set_set_a] :
( ~ ( member_set_a @ X2 @ A5 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A5 ) @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_1025_Diff__insert__absorb,axiom,
! [X2: a,A5: set_a] :
( ~ ( member_a @ X2 @ A5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A5 ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_1026_subset__Diff__insert,axiom,
! [A5: set_set_a,B4: set_set_a,X2: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ ( minus_5736297505244876581_set_a @ B4 @ ( insert_set_a @ X2 @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A5 @ ( minus_5736297505244876581_set_a @ B4 @ C2 ) )
& ~ ( member_set_a @ X2 @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_1027_subset__Diff__insert,axiom,
! [A5: set_a,B4: set_a,X2: a,C2: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( minus_minus_set_a @ B4 @ ( insert_a @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A5 @ ( minus_minus_set_a @ B4 @ C2 ) )
& ~ ( member_a @ X2 @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_1028_Un__Int__assoc__eq,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B4 ) @ C2 )
= ( inf_inf_set_a @ A5 @ ( sup_sup_set_a @ B4 @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A5 ) ) ).
% Un_Int_assoc_eq
thf(fact_1029_ivl__disj__int__two_I3_J,axiom,
! [L: a,M: a,U: a] :
( ( inf_inf_set_a @ ( set_or5139330845457685135Than_a @ L @ M ) @ ( set_or5139330845457685135Than_a @ M @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(3)
thf(fact_1030_Diff__partition,axiom,
! [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( sup_sup_set_a @ A5 @ ( minus_minus_set_a @ B4 @ A5 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_1031_Diff__subset__conv,axiom,
! [A5: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ B4 ) @ C2 )
= ( ord_less_eq_set_a @ A5 @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1032_ivl__disj__int__two_I6_J,axiom,
! [L: a,M: a,U: a] :
( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ L @ M ) @ ( set_or4472690218693186638Most_a @ M @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(6)
thf(fact_1033_ordering__top_Oaxioms_I1_J,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,Top: a] :
( ( ordering_top_a @ Less_eq2 @ Less @ Top )
=> ( ordering_a @ Less_eq2 @ Less ) ) ).
% ordering_top.axioms(1)
thf(fact_1034_boolean__algebra_Ocomplement__unique,axiom,
! [A: a > a > $o,X2: a > a > $o,Y3: a > a > $o] :
( ( ( inf_inf_a_a_o @ A @ X2 )
= bot_bot_a_a_o )
=> ( ( ( sup_sup_a_a_o @ A @ X2 )
= top_top_a_a_o )
=> ( ( ( inf_inf_a_a_o @ A @ Y3 )
= bot_bot_a_a_o )
=> ( ( ( sup_sup_a_a_o @ A @ Y3 )
= top_top_a_a_o )
=> ( X2 = Y3 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1035_boolean__algebra_Ocomplement__unique,axiom,
! [A: a > $o,X2: a > $o,Y3: a > $o] :
( ( ( inf_inf_a_o @ A @ X2 )
= bot_bot_a_o )
=> ( ( ( sup_sup_a_o @ A @ X2 )
= top_top_a_o )
=> ( ( ( inf_inf_a_o @ A @ Y3 )
= bot_bot_a_o )
=> ( ( ( sup_sup_a_o @ A @ Y3 )
= top_top_a_o )
=> ( X2 = Y3 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1036_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_a,X2: set_a,Y3: set_a] :
( ( ( inf_inf_set_a @ A @ X2 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A @ X2 )
= top_top_set_a )
=> ( ( ( inf_inf_set_a @ A @ Y3 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A @ Y3 )
= top_top_set_a )
=> ( X2 = Y3 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1037_Diff__single__insert,axiom,
! [A5: set_a,X2: a,B4: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B4 )
=> ( ord_less_eq_set_a @ A5 @ ( insert_a @ X2 @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_1038_subset__insert__iff,axiom,
! [A5: set_set_a,X2: set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ ( insert_set_a @ X2 @ B4 ) )
= ( ( ( member_set_a @ X2 @ A5 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B4 ) )
& ( ~ ( member_set_a @ X2 @ A5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1039_subset__insert__iff,axiom,
! [A5: set_a,X2: a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X2 @ B4 ) )
= ( ( ( member_a @ X2 @ A5 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B4 ) )
& ( ~ ( member_a @ X2 @ A5 )
=> ( ord_less_eq_set_a @ A5 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1040_ivl__disj__int__two_I7_J,axiom,
! [L: a,M: a,U: a] :
( ( inf_inf_set_a @ ( set_or5139330845457685135Than_a @ L @ M ) @ ( set_or672772299803893939Most_a @ M @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(7)
thf(fact_1041_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_1042_ivl__disj__int__one_I4_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or672772299803893939Most_a @ L @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(4)
thf(fact_1043_ivl__disj__int__one_I2_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_ord_lessThan_a @ L ) @ ( set_or5139330845457685135Than_a @ L @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(2)
thf(fact_1044_ivl__disj__int__two_I8_J,axiom,
! [L: a,M: a,U: a] :
( ( inf_inf_set_a @ ( set_or672772299803893939Most_a @ L @ M ) @ ( set_or4472690218693186638Most_a @ M @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(8)
thf(fact_1045_ivl__disj__int__two_I4_J,axiom,
! [L: a,M: a,U: a] :
( ( inf_inf_set_a @ ( set_or672772299803893939Most_a @ L @ M ) @ ( set_or5939364468397584554Than_a @ M @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(4)
thf(fact_1046_ivl__disj__int__two_I5_J,axiom,
! [L: a,M: a,U: a] :
( ( inf_inf_set_a @ ( set_or5939364468397584554Than_a @ L @ M ) @ ( set_or672772299803893939Most_a @ M @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(5)
thf(fact_1047_ivl__disj__int__two_I1_J,axiom,
! [L: a,M: a,U: a] :
( ( inf_inf_set_a @ ( set_or5939364468397584554Than_a @ L @ M ) @ ( set_or5139330845457685135Than_a @ M @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(1)
thf(fact_1048_ivl__disj__int__one_I8_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_or5139330845457685135Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(8)
thf(fact_1049_ivl__disj__int__one_I3_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or4472690218693186638Most_a @ L @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(3)
thf(fact_1050_ivl__disj__int__one_I7_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_or672772299803893939Most_a @ L @ U ) @ ( set_or8632414552788122084Than_a @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(7)
thf(fact_1051_ivl__disj__int__one_I1_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or5939364468397584554Than_a @ L @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(1)
thf(fact_1052_pairwise__alt,axiom,
( pairwise_a
= ( ^ [R2: a > a > $o,S: set_a] :
! [X3: a] :
( ( member_a @ X3 @ S )
=> ! [Y: a] :
( ( member_a @ Y @ ( minus_minus_set_a @ S @ ( insert_a @ X3 @ bot_bot_set_a ) ) )
=> ( R2 @ X3 @ Y ) ) ) ) ) ).
% pairwise_alt
thf(fact_1053_atLeastAtMost__def,axiom,
( set_or672772299803893939Most_a
= ( ^ [L3: a,U2: a] : ( inf_inf_set_a @ ( set_ord_atLeast_a @ L3 ) @ ( set_ord_atMost_a @ U2 ) ) ) ) ).
% atLeastAtMost_def
thf(fact_1054_ivl__disj__int__two_I2_J,axiom,
! [L: a,M: a,U: a] :
( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ L @ M ) @ ( set_or5939364468397584554Than_a @ M @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_two(2)
thf(fact_1055_atLeastLessThan__def,axiom,
( set_or5139330845457685135Than_a
= ( ^ [L3: a,U2: a] : ( inf_inf_set_a @ ( set_ord_atLeast_a @ L3 ) @ ( set_ord_lessThan_a @ U2 ) ) ) ) ).
% atLeastLessThan_def
thf(fact_1056_ivl__disj__int__one_I6_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_or5939364468397584554Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(6)
thf(fact_1057_ivl__disj__int__one_I5_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ L @ U ) @ ( set_or8632414552788122084Than_a @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(5)
thf(fact_1058_greaterThanAtMost__def,axiom,
( set_or4472690218693186638Most_a
= ( ^ [L3: a,U2: a] : ( inf_inf_set_a @ ( set_or8632414552788122084Than_a @ L3 ) @ ( set_ord_atMost_a @ U2 ) ) ) ) ).
% greaterThanAtMost_def
thf(fact_1059_greaterThanLessThan__eq,axiom,
( set_or5939364468397584554Than_a
= ( ^ [A2: a,B2: a] : ( inf_inf_set_a @ ( set_or8632414552788122084Than_a @ A2 ) @ ( set_ord_lessThan_a @ B2 ) ) ) ) ).
% greaterThanLessThan_eq
thf(fact_1060_greaterThanLessThan__def,axiom,
( set_or5939364468397584554Than_a
= ( ^ [L3: a,U2: a] : ( inf_inf_set_a @ ( set_or8632414552788122084Than_a @ L3 ) @ ( set_ord_lessThan_a @ U2 ) ) ) ) ).
% greaterThanLessThan_def
thf(fact_1061_Iio__Int__singleton,axiom,
! [X2: set_a,K: set_a] :
( ( ( ord_less_set_a @ X2 @ K )
=> ( ( inf_inf_set_set_a @ ( set_or5421148953861284865_set_a @ K ) @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) )
& ( ~ ( ord_less_set_a @ X2 @ K )
=> ( ( inf_inf_set_set_a @ ( set_or5421148953861284865_set_a @ K ) @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a ) ) ) ).
% Iio_Int_singleton
thf(fact_1062_Iio__Int__singleton,axiom,
! [X2: a,K: a] :
( ( ( ord_less_a @ X2 @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= ( insert_a @ X2 @ bot_bot_set_a ) ) )
& ( ~ ( ord_less_a @ X2 @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= bot_bot_set_a ) ) ) ).
% Iio_Int_singleton
thf(fact_1063_psubset__insert__iff,axiom,
! [A5: set_set_a,X2: set_a,B4: set_set_a] :
( ( ord_less_set_set_a @ A5 @ ( insert_set_a @ X2 @ B4 ) )
= ( ( ( member_set_a @ X2 @ B4 )
=> ( ord_less_set_set_a @ A5 @ B4 ) )
& ( ~ ( member_set_a @ X2 @ B4 )
=> ( ( ( member_set_a @ X2 @ A5 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B4 ) )
& ( ~ ( member_set_a @ X2 @ A5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1064_psubset__insert__iff,axiom,
! [A5: set_a,X2: a,B4: set_a] :
( ( ord_less_set_a @ A5 @ ( insert_a @ X2 @ B4 ) )
= ( ( ( member_a @ X2 @ B4 )
=> ( ord_less_set_a @ A5 @ B4 ) )
& ( ~ ( member_a @ X2 @ B4 )
=> ( ( ( member_a @ X2 @ A5 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B4 ) )
& ( ~ ( member_a @ X2 @ A5 )
=> ( ord_less_eq_set_a @ A5 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1065_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or5139330845457685135Than_a
= ( ^ [A2: a,B2: a] : ( minus_minus_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_1066_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
( set_or4472690218693186638Most_a
= ( ^ [A2: a,B2: a] : ( minus_minus_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_1067_Inf__empty,axiom,
( ( comple6135023378680113637_set_a @ bot_bot_set_set_a )
= top_top_set_a ) ).
% Inf_empty
thf(fact_1068_Inf__UNIV,axiom,
( ( comple6135023378680113637_set_a @ top_top_set_set_a )
= bot_bot_set_a ) ).
% Inf_UNIV
thf(fact_1069_Inf__UNIV,axiom,
( ( complete_Inf_Inf_a_o @ top_top_set_a_o )
= bot_bot_a_o ) ).
% Inf_UNIV
thf(fact_1070_less__eq__Inf__inter,axiom,
! [A5: set_a_a_o,B4: set_a_a_o] : ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ ( comple4872679785363069173_a_a_o @ A5 ) @ ( comple4872679785363069173_a_a_o @ B4 ) ) @ ( comple4872679785363069173_a_a_o @ ( inf_inf_set_a_a_o @ A5 @ B4 ) ) ) ).
% less_eq_Inf_inter
thf(fact_1071_less__eq__Inf__inter,axiom,
! [A5: set_set_a,B4: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B4 ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A5 @ B4 ) ) ) ).
% less_eq_Inf_inter
thf(fact_1072_Inter__empty,axiom,
( ( comple6135023378680113637_set_a @ bot_bot_set_set_a )
= top_top_set_a ) ).
% Inter_empty
thf(fact_1073_Inter__subset,axiom,
! [A5: set_set_a,B4: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A5 )
=> ( ord_less_eq_set_a @ X7 @ B4 ) )
=> ( ( A5 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ B4 ) ) ) ).
% Inter_subset
thf(fact_1074_Inter__anti__mono,axiom,
! [B4: set_set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B4 ) ) ) ).
% Inter_anti_mono
thf(fact_1075_Inter__greatest,axiom,
! [A5: set_set_a,C2: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A5 )
=> ( ord_less_eq_set_a @ C2 @ X7 ) )
=> ( ord_less_eq_set_a @ C2 @ ( comple6135023378680113637_set_a @ A5 ) ) ) ).
% Inter_greatest
thf(fact_1076_Inter__lower,axiom,
! [B4: set_a,A5: set_set_a] :
( ( member_set_a @ B4 @ A5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ B4 ) ) ).
% Inter_lower
thf(fact_1077_Inter__Un__distrib,axiom,
! [A5: set_set_a,B4: set_set_a] :
( ( comple6135023378680113637_set_a @ ( sup_sup_set_set_a @ A5 @ B4 ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B4 ) ) ) ).
% Inter_Un_distrib
thf(fact_1078_Inf__greatest,axiom,
! [A5: set_set_a,Z: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ord_less_eq_set_a @ Z @ X ) )
=> ( ord_less_eq_set_a @ Z @ ( comple6135023378680113637_set_a @ A5 ) ) ) ).
% Inf_greatest
thf(fact_1079_le__Inf__iff,axiom,
! [B: set_a,A5: set_set_a] :
( ( ord_less_eq_set_a @ B @ ( comple6135023378680113637_set_a @ A5 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( ord_less_eq_set_a @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_1080_Inf__lower2,axiom,
! [U: set_a,A5: set_set_a,V: set_a] :
( ( member_set_a @ U @ A5 )
=> ( ( ord_less_eq_set_a @ U @ V )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1081_Inf__lower,axiom,
! [X2: set_a,A5: set_set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ X2 ) ) ).
% Inf_lower
thf(fact_1082_Inf__mono,axiom,
! [B4: set_set_a,A5: set_set_a] :
( ! [B3: set_a] :
( ( member_set_a @ B3 @ B4 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
& ( ord_less_eq_set_a @ X4 @ B3 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B4 ) ) ) ).
% Inf_mono
thf(fact_1083_Inf__eqI,axiom,
! [A5: set_set_a,X2: set_a] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A5 )
=> ( ord_less_eq_set_a @ X2 @ I2 ) )
=> ( ! [Y2: set_a] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A5 )
=> ( ord_less_eq_set_a @ Y2 @ I3 ) )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ( comple6135023378680113637_set_a @ A5 )
= X2 ) ) ) ).
% Inf_eqI
thf(fact_1084_Inter__Un__subset,axiom,
! [A5: set_set_a,B4: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B4 ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A5 @ B4 ) ) ) ).
% Inter_Un_subset
thf(fact_1085_Inter__UNIV,axiom,
( ( comple6135023378680113637_set_a @ top_top_set_set_a )
= bot_bot_set_a ) ).
% Inter_UNIV
thf(fact_1086_Inf__less__eq,axiom,
! [A5: set_set_a,U: set_a] :
( ! [V2: set_a] :
( ( member_set_a @ V2 @ A5 )
=> ( ord_less_eq_set_a @ V2 @ U ) )
=> ( ( A5 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1087_Inf__superset__mono,axiom,
! [B4: set_set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B4 ) ) ) ).
% Inf_superset_mono
thf(fact_1088_Inf__union__distrib,axiom,
! [A5: set_set_a,B4: set_set_a] :
( ( comple6135023378680113637_set_a @ ( sup_sup_set_set_a @ A5 @ B4 ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B4 ) ) ) ).
% Inf_union_distrib
thf(fact_1089_Inf__sup__eq__top__iff,axiom,
! [B4: set_a_a_o,A: a > a > $o] :
( ( ( sup_sup_a_a_o @ ( comple4872679785363069173_a_a_o @ B4 ) @ A )
= top_top_a_a_o )
= ( ! [X3: a > a > $o] :
( ( member_a_a_o @ X3 @ B4 )
=> ( ( sup_sup_a_a_o @ X3 @ A )
= top_top_a_a_o ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_1090_Inf__sup__eq__top__iff,axiom,
! [B4: set_set_a,A: set_a] :
( ( ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ B4 ) @ A )
= top_top_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B4 )
=> ( ( sup_sup_set_a @ X3 @ A )
= top_top_set_a ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_1091_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2496817875450240012_set_a @ inf_inf_set_a @ top_top_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1092_ordering__top_Ointro,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,Top: a] :
( ( ordering_a @ Less_eq2 @ Less )
=> ( ( orderi2381714506568756172ioms_a @ Less_eq2 @ Top )
=> ( ordering_top_a @ Less_eq2 @ Less @ Top ) ) ) ).
% ordering_top.intro
thf(fact_1093_ordering__top__def,axiom,
( ordering_top_a
= ( ^ [Less_eq: a > a > $o,Less2: a > a > $o,Top2: a] :
( ( ordering_a @ Less_eq @ Less2 )
& ( orderi2381714506568756172ioms_a @ Less_eq @ Top2 ) ) ) ) ).
% ordering_top_def
thf(fact_1094_remove__def,axiom,
( remove_a
= ( ^ [X3: a,A4: set_a] : ( minus_minus_set_a @ A4 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_1095_less__eq__cInf__inter,axiom,
! [A5: set_set_a,B4: set_set_a] :
( ( condit8937546108433946286_set_a @ A5 )
=> ( ( condit8937546108433946286_set_a @ B4 )
=> ( ( ( inf_inf_set_set_a @ A5 @ B4 )
!= bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B4 ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A5 @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1096_chains__extend,axiom,
! [C: set_set_a,S2: set_set_a,Z: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S2 ) )
=> ( ( member_set_a @ Z @ S2 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ C )
=> ( ord_less_eq_set_a @ X @ Z ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z @ bot_bot_set_set_a ) @ C ) @ ( chains_a @ S2 ) ) ) ) ) ).
% chains_extend
thf(fact_1097_member__remove,axiom,
! [X2: set_a,Y3: set_a,A5: set_set_a] :
( ( member_set_a @ X2 @ ( remove_set_a @ Y3 @ A5 ) )
= ( ( member_set_a @ X2 @ A5 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_1098_member__remove,axiom,
! [X2: a,Y3: a,A5: set_a] :
( ( member_a @ X2 @ ( remove_a @ Y3 @ A5 ) )
= ( ( member_a @ X2 @ A5 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_1099_bdd__below_OI,axiom,
! [A5: set_o_a,M2: $o > a] :
( ! [X: $o > a] :
( ( member_o_a @ X @ A5 )
=> ( ord_less_eq_o_a @ M2 @ X ) )
=> ( condit8636343168986299771ow_o_a @ A5 ) ) ).
% bdd_below.I
thf(fact_1100_bdd__below_OI,axiom,
! [A5: set_set_a,M2: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ord_less_eq_set_a @ M2 @ X ) )
=> ( condit8937546108433946286_set_a @ A5 ) ) ).
% bdd_below.I
thf(fact_1101_bdd__below_OI,axiom,
! [A5: set_a,M2: a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ord_less_eq_a @ M2 @ X ) )
=> ( condit5901475214736682318elow_a @ A5 ) ) ).
% bdd_below.I
thf(fact_1102_bdd__belowI,axiom,
! [A5: set_o_a,M: $o > a] :
( ! [X: $o > a] :
( ( member_o_a @ X @ A5 )
=> ( ord_less_eq_o_a @ M @ X ) )
=> ( condit8636343168986299771ow_o_a @ A5 ) ) ).
% bdd_belowI
thf(fact_1103_bdd__belowI,axiom,
! [A5: set_set_a,M: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ord_less_eq_set_a @ M @ X ) )
=> ( condit8937546108433946286_set_a @ A5 ) ) ).
% bdd_belowI
thf(fact_1104_bdd__belowI,axiom,
! [A5: set_a,M: a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ord_less_eq_a @ M @ X ) )
=> ( condit5901475214736682318elow_a @ A5 ) ) ).
% bdd_belowI
thf(fact_1105_bdd__below__empty,axiom,
condit5901475214736682318elow_a @ bot_bot_set_a ).
% bdd_below_empty
thf(fact_1106_bdd__below__Icc,axiom,
! [A: a,B: a] : ( condit5901475214736682318elow_a @ ( set_or672772299803893939Most_a @ A @ B ) ) ).
% bdd_below_Icc
thf(fact_1107_bdd__below__Ico,axiom,
! [A: a,B: a] : ( condit5901475214736682318elow_a @ ( set_or5139330845457685135Than_a @ A @ B ) ) ).
% bdd_below_Ico
thf(fact_1108_bdd__below__Ioc,axiom,
! [A: a,B: a] : ( condit5901475214736682318elow_a @ ( set_or4472690218693186638Most_a @ A @ B ) ) ).
% bdd_below_Ioc
thf(fact_1109_bdd__below__Ioo,axiom,
! [A: a,B: a] : ( condit5901475214736682318elow_a @ ( set_or5939364468397584554Than_a @ A @ B ) ) ).
% bdd_below_Ioo
thf(fact_1110_bdd__below__Ici,axiom,
! [A: a] : ( condit5901475214736682318elow_a @ ( set_ord_atLeast_a @ A ) ) ).
% bdd_below_Ici
thf(fact_1111_bdd__below__Ioi,axiom,
! [A: a] : ( condit5901475214736682318elow_a @ ( set_or8632414552788122084Than_a @ A ) ) ).
% bdd_below_Ioi
thf(fact_1112_Zorn__Lemma2,axiom,
! [A5: set_set_a] :
( ! [X: set_set_a] :
( ( member_set_set_a @ X @ ( chains_a @ A5 ) )
=> ? [Xa: set_a] :
( ( member_set_a @ Xa @ A5 )
& ! [Xb: set_a] :
( ( member_set_a @ Xb @ X )
=> ( ord_less_eq_set_a @ Xb @ Xa ) ) ) )
=> ? [X: set_a] :
( ( member_set_a @ X @ A5 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A5 )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1113_chainsD,axiom,
! [C: set_set_a,S2: set_set_a,X2: set_a,Y3: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S2 ) )
=> ( ( member_set_a @ X2 @ C )
=> ( ( member_set_a @ Y3 @ C )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
| ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ) ).
% chainsD
thf(fact_1114_bdd__below_OE,axiom,
! [A5: set_o_a] :
( ( condit8636343168986299771ow_o_a @ A5 )
=> ~ ! [M3: $o > a] :
~ ! [X4: $o > a] :
( ( member_o_a @ X4 @ A5 )
=> ( ord_less_eq_o_a @ M3 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1115_bdd__below_OE,axiom,
! [A5: set_set_a] :
( ( condit8937546108433946286_set_a @ A5 )
=> ~ ! [M3: set_a] :
~ ! [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ( ord_less_eq_set_a @ M3 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1116_bdd__below_OE,axiom,
! [A5: set_a] :
( ( condit5901475214736682318elow_a @ A5 )
=> ~ ! [M3: a] :
~ ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( ord_less_eq_a @ M3 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1117_bdd__below_Ounfold,axiom,
( condit8636343168986299771ow_o_a
= ( ^ [A4: set_o_a] :
? [M4: $o > a] :
! [X3: $o > a] :
( ( member_o_a @ X3 @ A4 )
=> ( ord_less_eq_o_a @ M4 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1118_bdd__below_Ounfold,axiom,
( condit8937546108433946286_set_a
= ( ^ [A4: set_set_a] :
? [M4: set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
=> ( ord_less_eq_set_a @ M4 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1119_bdd__below_Ounfold,axiom,
( condit5901475214736682318elow_a
= ( ^ [A4: set_a] :
? [M4: a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( ord_less_eq_a @ M4 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1120_bdd__below__mono,axiom,
! [B4: set_a,A5: set_a] :
( ( condit5901475214736682318elow_a @ B4 )
=> ( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( condit5901475214736682318elow_a @ A5 ) ) ) ).
% bdd_below_mono
thf(fact_1121_cInf__lower2,axiom,
! [X2: set_a,X6: set_set_a,Y3: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( condit8937546108433946286_set_a @ X6 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X6 ) @ Y3 ) ) ) ) ).
% cInf_lower2
thf(fact_1122_cInf__lower,axiom,
! [X2: set_a,X6: set_set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ( condit8937546108433946286_set_a @ X6 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X6 ) @ X2 ) ) ) ).
% cInf_lower
thf(fact_1123_bdd__below__Int1,axiom,
! [A5: set_a,B4: set_a] :
( ( condit5901475214736682318elow_a @ A5 )
=> ( condit5901475214736682318elow_a @ ( inf_inf_set_a @ A5 @ B4 ) ) ) ).
% bdd_below_Int1
thf(fact_1124_bdd__below__Int2,axiom,
! [B4: set_a,A5: set_a] :
( ( condit5901475214736682318elow_a @ B4 )
=> ( condit5901475214736682318elow_a @ ( inf_inf_set_a @ A5 @ B4 ) ) ) ).
% bdd_below_Int2
thf(fact_1125_le__cInf__iff,axiom,
! [S2: set_set_a,A: set_a] :
( ( S2 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ S2 )
=> ( ( ord_less_eq_set_a @ A @ ( comple6135023378680113637_set_a @ S2 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ S2 )
=> ( ord_less_eq_set_a @ A @ X3 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1126_cInf__mono,axiom,
! [B4: set_set_a,A5: set_set_a] :
( ( B4 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ A5 )
=> ( ! [B3: set_a] :
( ( member_set_a @ B3 @ B4 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
& ( ord_less_eq_set_a @ X4 @ B3 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_1127_cInf__superset__mono,axiom,
! [A5: set_set_a,B4: set_set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ B4 )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B4 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ B4 ) @ ( comple6135023378680113637_set_a @ A5 ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1128_cInf__insert,axiom,
! [X6: set_set_a,A: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ X6 )
=> ( ( comple6135023378680113637_set_a @ ( insert_set_a @ A @ X6 ) )
= ( inf_inf_set_a @ A @ ( comple6135023378680113637_set_a @ X6 ) ) ) ) ) ).
% cInf_insert
thf(fact_1129_cInf__insert__If,axiom,
! [X6: set_set_a,A: set_a] :
( ( condit8937546108433946286_set_a @ X6 )
=> ( ( ( X6 = bot_bot_set_set_a )
=> ( ( comple6135023378680113637_set_a @ ( insert_set_a @ A @ X6 ) )
= A ) )
& ( ( X6 != bot_bot_set_set_a )
=> ( ( comple6135023378680113637_set_a @ ( insert_set_a @ A @ X6 ) )
= ( inf_inf_set_a @ A @ ( comple6135023378680113637_set_a @ X6 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_1130_cInf__union__distrib,axiom,
! [A5: set_set_a,B4: set_set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ A5 )
=> ( ( B4 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ B4 )
=> ( ( comple6135023378680113637_set_a @ ( sup_sup_set_set_a @ A5 @ B4 ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A5 ) @ ( comple6135023378680113637_set_a @ B4 ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_1131_cINF__union,axiom,
! [A5: set_set_a,F: set_a > set_a,B4: set_set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A5 ) )
=> ( ( B4 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ B4 ) )
=> ( ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ ( sup_sup_set_set_a @ A5 @ B4 ) ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A5 ) ) @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_1132_cINF__union,axiom,
! [A5: set_a,F: a > set_a,B4: set_a] :
( ( A5 != bot_bot_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A5 ) )
=> ( ( B4 != bot_bot_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ B4 ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ ( sup_sup_set_a @ A5 @ B4 ) ) )
= ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A5 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_1133_cINF__insert,axiom,
! [A5: set_set_a,F: set_a > set_a,A: set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A5 ) )
=> ( ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ ( insert_set_a @ A @ A5 ) ) )
= ( inf_inf_set_a @ ( F @ A ) @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A5 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_1134_cINF__insert,axiom,
! [A5: set_a,F: a > set_a,A: a] :
( ( A5 != bot_bot_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A5 ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ ( insert_a @ A @ A5 ) ) )
= ( inf_inf_set_a @ ( F @ A ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A5 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_1135_image__eqI,axiom,
! [B: set_a,F: a > set_a,X2: a,A5: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A5 )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_1136_image__eqI,axiom,
! [B: a,F: set_a > a,X2: set_a,A5: set_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_a @ X2 @ A5 )
=> ( member_a @ B @ ( image_set_a_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_1137_image__eqI,axiom,
! [B: set_a,F: set_a > set_a,X2: set_a,A5: set_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_a @ X2 @ A5 )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_1138_image__eqI,axiom,
! [B: a,F: a > a,X2: a,A5: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A5 )
=> ( member_a @ B @ ( image_a_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_1139_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_1140_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_1141_empty__is__image,axiom,
! [F: set_a > set_a,A5: set_set_a] :
( ( bot_bot_set_set_a
= ( image_set_a_set_a @ F @ A5 ) )
= ( A5 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_1142_empty__is__image,axiom,
! [F: a > a,A5: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A5 ) )
= ( A5 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_1143_image__is__empty,axiom,
! [F: set_a > set_a,A5: set_set_a] :
( ( ( image_set_a_set_a @ F @ A5 )
= bot_bot_set_set_a )
= ( A5 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_1144_image__is__empty,axiom,
! [F: a > a,A5: set_a] :
( ( ( image_a_a @ F @ A5 )
= bot_bot_set_a )
= ( A5 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1145_image__insert,axiom,
! [F: set_a > set_a,A: set_a,B4: set_set_a] :
( ( image_set_a_set_a @ F @ ( insert_set_a @ A @ B4 ) )
= ( insert_set_a @ ( F @ A ) @ ( image_set_a_set_a @ F @ B4 ) ) ) ).
% image_insert
thf(fact_1146_image__insert,axiom,
! [F: a > a,A: a,B4: set_a] :
( ( image_a_a @ F @ ( insert_a @ A @ B4 ) )
= ( insert_a @ ( F @ A ) @ ( image_a_a @ F @ B4 ) ) ) ).
% image_insert
thf(fact_1147_insert__image,axiom,
! [X2: a,A5: set_a,F: a > a] :
( ( member_a @ X2 @ A5 )
=> ( ( insert_a @ ( F @ X2 ) @ ( image_a_a @ F @ A5 ) )
= ( image_a_a @ F @ A5 ) ) ) ).
% insert_image
thf(fact_1148_insert__image,axiom,
! [X2: set_a,A5: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( ( insert_set_a @ ( F @ X2 ) @ ( image_set_a_set_a @ F @ A5 ) )
= ( image_set_a_set_a @ F @ A5 ) ) ) ).
% insert_image
thf(fact_1149_insert__image,axiom,
! [X2: set_a,A5: set_set_a,F: set_a > a] :
( ( member_set_a @ X2 @ A5 )
=> ( ( insert_a @ ( F @ X2 ) @ ( image_set_a_a @ F @ A5 ) )
= ( image_set_a_a @ F @ A5 ) ) ) ).
% insert_image
thf(fact_1150_bdd__belowI2,axiom,
! [A5: set_a,M: $o > a,F: a > $o > a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ord_less_eq_o_a @ M @ ( F @ X ) ) )
=> ( condit8636343168986299771ow_o_a @ ( image_a_o_a @ F @ A5 ) ) ) ).
% bdd_belowI2
thf(fact_1151_bdd__belowI2,axiom,
! [A5: set_set_a,M: $o > a,F: set_a > $o > a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ord_less_eq_o_a @ M @ ( F @ X ) ) )
=> ( condit8636343168986299771ow_o_a @ ( image_set_a_o_a @ F @ A5 ) ) ) ).
% bdd_belowI2
thf(fact_1152_bdd__belowI2,axiom,
! [A5: set_a,M: set_a,F: a > set_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ord_less_eq_set_a @ M @ ( F @ X ) ) )
=> ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A5 ) ) ) ).
% bdd_belowI2
thf(fact_1153_bdd__belowI2,axiom,
! [A5: set_set_a,M: set_a,F: set_a > set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ord_less_eq_set_a @ M @ ( F @ X ) ) )
=> ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A5 ) ) ) ).
% bdd_belowI2
thf(fact_1154_bdd__belowI2,axiom,
! [A5: set_set_a,M: a,F: set_a > a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ord_less_eq_a @ M @ ( F @ X ) ) )
=> ( condit5901475214736682318elow_a @ ( image_set_a_a @ F @ A5 ) ) ) ).
% bdd_belowI2
thf(fact_1155_bdd__belowI2,axiom,
! [A5: set_a,M: a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ord_less_eq_a @ M @ ( F @ X ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A5 ) ) ) ).
% bdd_belowI2
thf(fact_1156_bdd__below_OI2,axiom,
! [A5: set_a,M2: $o > a,F: a > $o > a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ord_less_eq_o_a @ M2 @ ( F @ X ) ) )
=> ( condit8636343168986299771ow_o_a @ ( image_a_o_a @ F @ A5 ) ) ) ).
% bdd_below.I2
thf(fact_1157_bdd__below_OI2,axiom,
! [A5: set_set_a,M2: $o > a,F: set_a > $o > a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ord_less_eq_o_a @ M2 @ ( F @ X ) ) )
=> ( condit8636343168986299771ow_o_a @ ( image_set_a_o_a @ F @ A5 ) ) ) ).
% bdd_below.I2
thf(fact_1158_bdd__below_OI2,axiom,
! [A5: set_a,M2: set_a,F: a > set_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ord_less_eq_set_a @ M2 @ ( F @ X ) ) )
=> ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A5 ) ) ) ).
% bdd_below.I2
thf(fact_1159_bdd__below_OI2,axiom,
! [A5: set_set_a,M2: set_a,F: set_a > set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ord_less_eq_set_a @ M2 @ ( F @ X ) ) )
=> ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A5 ) ) ) ).
% bdd_below.I2
thf(fact_1160_bdd__below_OI2,axiom,
! [A5: set_set_a,M2: a,F: set_a > a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ord_less_eq_a @ M2 @ ( F @ X ) ) )
=> ( condit5901475214736682318elow_a @ ( image_set_a_a @ F @ A5 ) ) ) ).
% bdd_below.I2
thf(fact_1161_bdd__below_OI2,axiom,
! [A5: set_a,M2: a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ord_less_eq_a @ M2 @ ( F @ X ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A5 ) ) ) ).
% bdd_below.I2
thf(fact_1162_INF__eq,axiom,
! [A5: set_a,B4: set_a,G2: a > set_a,F: a > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A5 )
=> ? [X4: a] :
( ( member_a @ X4 @ B4 )
& ( ord_less_eq_set_a @ ( G2 @ X4 ) @ ( F @ I2 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B4 )
=> ? [X4: a] :
( ( member_a @ X4 @ A5 )
& ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G2 @ J2 ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A5 ) )
= ( comple6135023378680113637_set_a @ ( image_a_set_a @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_1163_INF__eq,axiom,
! [A5: set_a,B4: set_set_a,G2: set_a > set_a,F: a > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A5 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ B4 )
& ( ord_less_eq_set_a @ ( G2 @ X4 ) @ ( F @ I2 ) ) ) )
=> ( ! [J2: set_a] :
( ( member_set_a @ J2 @ B4 )
=> ? [X4: a] :
( ( member_a @ X4 @ A5 )
& ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G2 @ J2 ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A5 ) )
= ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_1164_INF__eq,axiom,
! [A5: set_set_a,B4: set_a,G2: a > set_a,F: set_a > set_a] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A5 )
=> ? [X4: a] :
( ( member_a @ X4 @ B4 )
& ( ord_less_eq_set_a @ ( G2 @ X4 ) @ ( F @ I2 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B4 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
& ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G2 @ J2 ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A5 ) )
= ( comple6135023378680113637_set_a @ ( image_a_set_a @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_1165_INF__eq,axiom,
! [A5: set_set_a,B4: set_set_a,G2: set_a > set_a,F: set_a > set_a] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A5 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ B4 )
& ( ord_less_eq_set_a @ ( G2 @ X4 ) @ ( F @ I2 ) ) ) )
=> ( ! [J2: set_a] :
( ( member_set_a @ J2 @ B4 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
& ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G2 @ J2 ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A5 ) )
= ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_1166_INF__eq__const,axiom,
! [I4: set_set_a,F: set_a > set_a,X2: set_a] :
( ( I4 != bot_bot_set_set_a )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I4 )
=> ( ( F @ I2 )
= X2 ) )
=> ( ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ I4 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_1167_range__subsetD,axiom,
! [F: set_a > set_a,B4: set_set_a,I: set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ top_top_set_set_a ) @ B4 )
=> ( member_set_a @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_1168_range__subsetD,axiom,
! [F: a > set_a,B4: set_set_a,I: a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ top_top_set_a ) @ B4 )
=> ( member_set_a @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_1169_range__subsetD,axiom,
! [F: a > a,B4: set_a,I: a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ top_top_set_a ) @ B4 )
=> ( member_a @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_1170_image__Int__subset,axiom,
! [F: set_a > set_a,A5: set_set_a,B4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ ( inf_inf_set_set_a @ A5 @ B4 ) ) @ ( inf_inf_set_set_a @ ( image_set_a_set_a @ F @ A5 ) @ ( image_set_a_set_a @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_1171_image__Int__subset,axiom,
! [F: a > a,A5: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A5 @ B4 ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A5 ) @ ( image_a_a @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_1172_image__diff__subset,axiom,
! [F: set_a > set_a,A5: set_set_a,B4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_set_a_set_a @ F @ A5 ) @ ( image_set_a_set_a @ F @ B4 ) ) @ ( image_set_a_set_a @ F @ ( minus_5736297505244876581_set_a @ A5 @ B4 ) ) ) ).
% image_diff_subset
thf(fact_1173_image__diff__subset,axiom,
! [F: a > a,A5: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A5 ) @ ( image_a_a @ F @ B4 ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A5 @ B4 ) ) ) ).
% image_diff_subset
thf(fact_1174_image__Un,axiom,
! [F: set_a > set_a,A5: set_set_a,B4: set_set_a] :
( ( image_set_a_set_a @ F @ ( sup_sup_set_set_a @ A5 @ B4 ) )
= ( sup_sup_set_set_a @ ( image_set_a_set_a @ F @ A5 ) @ ( image_set_a_set_a @ F @ B4 ) ) ) ).
% image_Un
thf(fact_1175_image__Un,axiom,
! [F: a > a,A5: set_a,B4: set_a] :
( ( image_a_a @ F @ ( sup_sup_set_a @ A5 @ B4 ) )
= ( sup_sup_set_a @ ( image_a_a @ F @ A5 ) @ ( image_a_a @ F @ B4 ) ) ) ).
% image_Un
thf(fact_1176_image__mono,axiom,
! [A5: set_set_a,B4: set_set_a,F: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B4 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A5 ) @ ( image_set_a_set_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_1177_image__mono,axiom,
! [A5: set_a,B4: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A5 ) @ ( image_a_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_1178_image__subsetI,axiom,
! [A5: set_a,F: a > set_a,B4: set_set_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_set_a @ ( F @ X ) @ B4 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A5 ) @ B4 ) ) ).
% image_subsetI
thf(fact_1179_image__subsetI,axiom,
! [A5: set_set_a,F: set_a > set_a,B4: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( member_set_a @ ( F @ X ) @ B4 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A5 ) @ B4 ) ) ).
% image_subsetI
thf(fact_1180_image__subsetI,axiom,
! [A5: set_set_a,F: set_a > a,B4: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( member_a @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A5 ) @ B4 ) ) ).
% image_subsetI
thf(fact_1181_image__subsetI,axiom,
! [A5: set_a,F: a > a,B4: set_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A5 ) @ B4 ) ) ).
% image_subsetI
thf(fact_1182_subset__imageE,axiom,
! [B4: set_set_a,F: set_a > set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_set_a_set_a @ F @ A5 ) )
=> ~ ! [C6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C6 @ A5 )
=> ( B4
!= ( image_set_a_set_a @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_1183_subset__imageE,axiom,
! [B4: set_a,F: a > a,A5: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A5 ) )
=> ~ ! [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A5 )
=> ( B4
!= ( image_a_a @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_1184_image__subset__iff,axiom,
! [F: set_a > set_a,A5: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A5 ) @ B4 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( member_set_a @ ( F @ X3 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_1185_image__subset__iff,axiom,
! [F: a > a,A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A5 ) @ B4 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ ( F @ X3 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_1186_subset__image__iff,axiom,
! [B4: set_set_a,F: set_a > set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_set_a_set_a @ F @ A5 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A5 )
& ( B4
= ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1187_subset__image__iff,axiom,
! [B4: set_a,F: a > a,A5: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A5 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A5 )
& ( B4
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1188_sup__Inf,axiom,
! [A: set_a,B4: set_set_a] :
( ( sup_sup_set_a @ A @ ( comple6135023378680113637_set_a @ B4 ) )
= ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ ( sup_sup_set_a @ A ) @ B4 ) ) ) ).
% sup_Inf
thf(fact_1189_sup__Inf,axiom,
! [A: a > a > $o,B4: set_a_a_o] :
( ( sup_sup_a_a_o @ A @ ( comple4872679785363069173_a_a_o @ B4 ) )
= ( comple4872679785363069173_a_a_o @ ( image_a_a_o_a_a_o @ ( sup_sup_a_a_o @ A ) @ B4 ) ) ) ).
% sup_Inf
thf(fact_1190_pairwise__imageI,axiom,
! [A5: set_set_a,F: set_a > set_a,P: set_a > set_a > $o] :
( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ( member_set_a @ Y2 @ A5 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_set_a @ P @ ( image_set_a_set_a @ F @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_1191_pairwise__imageI,axiom,
! [A5: set_a,F: a > a,P: a > a > $o] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ A5 )
=> ( ( member_a @ Y2 @ A5 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_a @ P @ ( image_a_a @ F @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_1192_pairwise__imageI,axiom,
! [A5: set_set_a,F: set_a > a,P: a > a > $o] :
( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ( member_set_a @ Y2 @ A5 )
=> ( ( X != Y2 )
=> ( ( ( F @ X )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_a @ P @ ( image_set_a_a @ F @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_1193_imageI,axiom,
! [X2: a,A5: set_a,F: a > set_a] :
( ( member_a @ X2 @ A5 )
=> ( member_set_a @ ( F @ X2 ) @ ( image_a_set_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_1194_imageI,axiom,
! [X2: set_a,A5: set_set_a,F: set_a > a] :
( ( member_set_a @ X2 @ A5 )
=> ( member_a @ ( F @ X2 ) @ ( image_set_a_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_1195_imageI,axiom,
! [X2: set_a,A5: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( member_set_a @ ( F @ X2 ) @ ( image_set_a_set_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_1196_imageI,axiom,
! [X2: a,A5: set_a,F: a > a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_1197_image__iff,axiom,
! [Z: a,F: a > a,A5: set_a] :
( ( member_a @ Z @ ( image_a_a @ F @ A5 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A5 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_1198_image__iff,axiom,
! [Z: set_a,F: set_a > set_a,A5: set_set_a] :
( ( member_set_a @ Z @ ( image_set_a_set_a @ F @ A5 ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_1199_bex__imageD,axiom,
! [F: a > a,A5: set_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A5 ) )
& ( P @ X4 ) )
=> ? [X: a] :
( ( member_a @ X @ A5 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_1200_bex__imageD,axiom,
! [F: set_a > set_a,A5: set_set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( image_set_a_set_a @ F @ A5 ) )
& ( P @ X4 ) )
=> ? [X: set_a] :
( ( member_set_a @ X @ A5 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_1201_image__cong,axiom,
! [M2: set_a,N2: set_a,F: a > a,G2: a > a] :
( ( M2 = N2 )
=> ( ! [X: a] :
( ( member_a @ X @ N2 )
=> ( ( F @ X )
= ( G2 @ X ) ) )
=> ( ( image_a_a @ F @ M2 )
= ( image_a_a @ G2 @ N2 ) ) ) ) ).
% image_cong
thf(fact_1202_image__cong,axiom,
! [M2: set_set_a,N2: set_set_a,F: set_a > set_a,G2: set_a > set_a] :
( ( M2 = N2 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ N2 )
=> ( ( F @ X )
= ( G2 @ X ) ) )
=> ( ( image_set_a_set_a @ F @ M2 )
= ( image_set_a_set_a @ G2 @ N2 ) ) ) ) ).
% image_cong
thf(fact_1203_ball__imageD,axiom,
! [F: a > a,A5: set_a,P: a > $o] :
( ! [X: a] :
( ( member_a @ X @ ( image_a_a @ F @ A5 ) )
=> ( P @ X ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1204_ball__imageD,axiom,
! [F: set_a > set_a,A5: set_set_a,P: set_a > $o] :
( ! [X: set_a] :
( ( member_set_a @ X @ ( image_set_a_set_a @ F @ A5 ) )
=> ( P @ X ) )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1205_rev__image__eqI,axiom,
! [X2: a,A5: set_a,B: set_a,F: a > set_a] :
( ( member_a @ X2 @ A5 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1206_rev__image__eqI,axiom,
! [X2: set_a,A5: set_set_a,B: a,F: set_a > a] :
( ( member_set_a @ X2 @ A5 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_set_a_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1207_rev__image__eqI,axiom,
! [X2: set_a,A5: set_set_a,B: set_a,F: set_a > set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1208_rev__image__eqI,axiom,
! [X2: a,A5: set_a,B: a,F: a > a] :
( ( member_a @ X2 @ A5 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_a_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1209_range__eqI,axiom,
! [B: set_a,F: set_a > set_a,X2: set_a] :
( ( B
= ( F @ X2 ) )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ top_top_set_set_a ) ) ) ).
% range_eqI
thf(fact_1210_range__eqI,axiom,
! [B: a,F: a > a,X2: a] :
( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_a_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1211_range__eqI,axiom,
! [B: set_a,F: a > set_a,X2: a] :
( ( B
= ( F @ X2 ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1212_rangeI,axiom,
! [F: set_a > set_a,X2: set_a] : ( member_set_a @ ( F @ X2 ) @ ( image_set_a_set_a @ F @ top_top_set_set_a ) ) ).
% rangeI
thf(fact_1213_rangeI,axiom,
! [F: a > a,X2: a] : ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_1214_rangeI,axiom,
! [F: a > set_a,X2: a] : ( member_set_a @ ( F @ X2 ) @ ( image_a_set_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_1215_the__elem__image__unique,axiom,
! [A5: set_set_a,F: set_a > set_a,X2: set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ! [Y2: set_a] :
( ( member_set_a @ Y2 @ A5 )
=> ( ( F @ Y2 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_a @ ( image_set_a_set_a @ F @ A5 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1216_the__elem__image__unique,axiom,
! [A5: set_a,F: a > a,X2: a] :
( ( A5 != bot_bot_set_a )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ A5 )
=> ( ( F @ Y2 )
= ( F @ X2 ) ) )
=> ( ( the_elem_a @ ( image_a_a @ F @ A5 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1217_INF__eq__iff,axiom,
! [I4: set_set_a,F: set_a > set_a,C: set_a] :
( ( I4 != bot_bot_set_set_a )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I4 )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ C ) )
=> ( ( ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ I4 ) )
= C )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1218_INF__eq__iff,axiom,
! [I4: set_a,F: a > set_a,C: set_a] :
( ( I4 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I4 )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ C ) )
=> ( ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ I4 ) )
= C )
= ( ! [X3: a] :
( ( member_a @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1219_cINF__greatest,axiom,
! [A5: set_set_a,M: set_a,F: set_a > set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ord_less_eq_set_a @ M @ ( F @ X ) ) )
=> ( ord_less_eq_set_a @ M @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A5 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1220_cINF__greatest,axiom,
! [A5: set_a,M: set_a,F: a > set_a] :
( ( A5 != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ord_less_eq_set_a @ M @ ( F @ X ) ) )
=> ( ord_less_eq_set_a @ M @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A5 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1221_range__eq__singletonD,axiom,
! [F: set_a > set_a,A: set_a,X2: set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1222_range__eq__singletonD,axiom,
! [F: a > a,A: a,X2: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= ( insert_a @ A @ bot_bot_set_a ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1223_in__image__insert__iff,axiom,
! [B4: set_set_set_a,X2: set_a,A5: set_set_a] :
( ! [C6: set_set_a] :
( ( member_set_set_a @ C6 @ B4 )
=> ~ ( member_set_a @ X2 @ C6 ) )
=> ( ( member_set_set_a @ A5 @ ( image_1042221919965026181_set_a @ ( insert_set_a @ X2 ) @ B4 ) )
= ( ( member_set_a @ X2 @ A5 )
& ( member_set_set_a @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1224_in__image__insert__iff,axiom,
! [B4: set_set_a,X2: a,A5: set_a] :
( ! [C6: set_a] :
( ( member_set_a @ C6 @ B4 )
=> ~ ( member_a @ X2 @ C6 ) )
=> ( ( member_set_a @ A5 @ ( image_set_a_set_a @ ( insert_a @ X2 ) @ B4 ) )
= ( ( member_a @ X2 @ A5 )
& ( member_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1225_cINF__lower,axiom,
! [F: a > set_a,A5: set_a,X2: a] :
( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A5 ) )
=> ( ( member_a @ X2 @ A5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A5 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_1226_cINF__lower,axiom,
! [F: set_a > set_a,A5: set_set_a,X2: set_a] :
( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A5 ) )
=> ( ( member_set_a @ X2 @ A5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A5 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_1227_cINF__lower2,axiom,
! [F: a > set_a,A5: set_a,X2: a,U: set_a] :
( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A5 ) )
=> ( ( member_a @ X2 @ A5 )
=> ( ( ord_less_eq_set_a @ ( F @ X2 ) @ U )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A5 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_1228_cINF__lower2,axiom,
! [F: set_a > set_a,A5: set_set_a,X2: set_a,U: set_a] :
( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A5 ) )
=> ( ( member_set_a @ X2 @ A5 )
=> ( ( ord_less_eq_set_a @ ( F @ X2 ) @ U )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A5 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_1229_cINF__mono,axiom,
! [B4: set_set_a,F: set_a > set_a,A5: set_set_a,G2: set_a > set_a] :
( ( B4 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A5 ) )
=> ( ! [M5: set_a] :
( ( member_set_a @ M5 @ B4 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
& ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G2 @ M5 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A5 ) ) @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ G2 @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_1230_cINF__mono,axiom,
! [B4: set_a,F: set_a > set_a,A5: set_set_a,G2: a > set_a] :
( ( B4 != bot_bot_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A5 ) )
=> ( ! [M5: a] :
( ( member_a @ M5 @ B4 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
& ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G2 @ M5 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A5 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G2 @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_1231_le__cINF__iff,axiom,
! [A5: set_set_a,F: set_a > set_a,U: set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A5 ) )
=> ( ( ord_less_eq_set_a @ U @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A5 ) ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( ord_less_eq_set_a @ U @ ( F @ X3 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_1232_le__cINF__iff,axiom,
! [A5: set_a,F: a > set_a,U: set_a] :
( ( A5 != bot_bot_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A5 ) )
=> ( ( ord_less_eq_set_a @ U @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A5 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( ord_less_eq_set_a @ U @ ( F @ X3 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_1233_cINF__superset__mono,axiom,
! [A5: set_set_a,G2: set_a > set_a,B4: set_set_a,F: set_a > set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ G2 @ B4 ) )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B4 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ B4 )
=> ( ord_less_eq_set_a @ ( G2 @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ G2 @ B4 ) ) @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A5 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1234_cINF__superset__mono,axiom,
! [A5: set_a,G2: a > set_a,B4: set_a,F: a > set_a] :
( ( A5 != bot_bot_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_a_set_a @ G2 @ B4 ) )
=> ( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ! [X: a] :
( ( member_a @ X @ B4 )
=> ( ord_less_eq_set_a @ ( G2 @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G2 @ B4 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A5 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1235_all__subset__image,axiom,
! [F: set_a > set_a,A5: set_set_a,P: set_set_a > $o] :
( ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ ( image_set_a_set_a @ F @ A5 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
=> ( P @ ( image_set_a_set_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1236_all__subset__image,axiom,
! [F: a > a,A5: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A5 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1237_subset_Ochain__extend,axiom,
! [A5: set_set_a,C2: set_set_a,Z: set_a] :
( ( pred_chain_set_a @ A5 @ ord_less_set_a @ C2 )
=> ( ( member_set_a @ Z @ A5 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ C2 )
=> ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ X
@ Z ) )
=> ( pred_chain_set_a @ A5 @ ord_less_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z @ bot_bot_set_set_a ) @ C2 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_1238_sup2CI,axiom,
! [B4: a > a > $o,X2: a,Y3: a,A5: a > a > $o] :
( ( ~ ( B4 @ X2 @ Y3 )
=> ( A5 @ X2 @ Y3 ) )
=> ( sup_sup_a_a_o @ A5 @ B4 @ X2 @ Y3 ) ) ).
% sup2CI
thf(fact_1239_subset_Ochain__empty,axiom,
! [A5: set_set_a] : ( pred_chain_set_a @ A5 @ ord_less_set_a @ bot_bot_set_set_a ) ).
% subset.chain_empty
thf(fact_1240_sup2E,axiom,
! [A5: a > a > $o,B4: a > a > $o,X2: a,Y3: a] :
( ( sup_sup_a_a_o @ A5 @ B4 @ X2 @ Y3 )
=> ( ~ ( A5 @ X2 @ Y3 )
=> ( B4 @ X2 @ Y3 ) ) ) ).
% sup2E
thf(fact_1241_sup2I1,axiom,
! [A5: a > a > $o,X2: a,Y3: a,B4: a > a > $o] :
( ( A5 @ X2 @ Y3 )
=> ( sup_sup_a_a_o @ A5 @ B4 @ X2 @ Y3 ) ) ).
% sup2I1
thf(fact_1242_sup2I2,axiom,
! [B4: a > a > $o,X2: a,Y3: a,A5: a > a > $o] :
( ( B4 @ X2 @ Y3 )
=> ( sup_sup_a_a_o @ A5 @ B4 @ X2 @ Y3 ) ) ).
% sup2I2
thf(fact_1243_pred__on_Ochain__empty,axiom,
! [A5: set_a,P: a > a > $o] : ( pred_chain_a @ A5 @ P @ bot_bot_set_a ) ).
% pred_on.chain_empty
thf(fact_1244_pred__on_Ochain__total,axiom,
! [A5: set_set_a,P: set_a > set_a > $o,C2: set_set_a,X2: set_a,Y3: set_a] :
( ( pred_chain_set_a @ A5 @ P @ C2 )
=> ( ( member_set_a @ X2 @ C2 )
=> ( ( member_set_a @ Y3 @ C2 )
=> ( ( sup_su8131220024717662786et_a_o @ P
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ X2
@ Y3 )
| ( sup_su8131220024717662786et_a_o @ P
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ Y3
@ X2 ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_1245_pred__on_Ochain__total,axiom,
! [A5: set_a,P: a > a > $o,C2: set_a,X2: a,Y3: a] :
( ( pred_chain_a @ A5 @ P @ C2 )
=> ( ( member_a @ X2 @ C2 )
=> ( ( member_a @ Y3 @ C2 )
=> ( ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ X2
@ Y3 )
| ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ Y3
@ X2 ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_1246_pred__on_Ochain__def,axiom,
( pred_chain_a
= ( ^ [A4: set_a,P2: a > a > $o,C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A4 )
& ! [X3: a] :
( ( member_a @ X3 @ C5 )
=> ! [Y: a] :
( ( member_a @ Y @ C5 )
=> ( ( sup_sup_a_a_o @ P2
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ X3
@ Y )
| ( sup_sup_a_a_o @ P2
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ Y
@ X3 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_1247_pred__on_OchainI,axiom,
! [C2: set_set_a,A5: set_set_a,P: set_a > set_a > $o] :
( ( ord_le3724670747650509150_set_a @ C2 @ A5 )
=> ( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ C2 )
=> ( ( member_set_a @ Y2 @ C2 )
=> ( ( sup_su8131220024717662786et_a_o @ P
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ X
@ Y2 )
| ( sup_su8131220024717662786et_a_o @ P
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ Y2
@ X ) ) ) )
=> ( pred_chain_set_a @ A5 @ P @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_1248_pred__on_OchainI,axiom,
! [C2: set_a,A5: set_a,P: a > a > $o] :
( ( ord_less_eq_set_a @ C2 @ A5 )
=> ( ! [X: a,Y2: a] :
( ( member_a @ X @ C2 )
=> ( ( member_a @ Y2 @ C2 )
=> ( ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ X
@ Y2 )
| ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ Y2
@ X ) ) ) )
=> ( pred_chain_a @ A5 @ P @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_1249_subset_OchainI,axiom,
! [C2: set_set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A5 )
=> ( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ C2 )
=> ( ( member_set_a @ Y2 @ C2 )
=> ( ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ X
@ Y2 )
| ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ Y2
@ X ) ) ) )
=> ( pred_chain_set_a @ A5 @ ord_less_set_a @ C2 ) ) ) ).
% subset.chainI
thf(fact_1250_subset_Ochain__def,axiom,
! [A5: set_set_a,C2: set_set_a] :
( ( pred_chain_set_a @ A5 @ ord_less_set_a @ C2 )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A5 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ C2 )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ C2 )
=> ( ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ X3
@ Y )
| ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ Y
@ X3 ) ) ) ) ) ) ).
% subset.chain_def
thf(fact_1251_subset_Ochain__total,axiom,
! [A5: set_set_a,C2: set_set_a,X2: set_a,Y3: set_a] :
( ( pred_chain_set_a @ A5 @ ord_less_set_a @ C2 )
=> ( ( member_set_a @ X2 @ C2 )
=> ( ( member_set_a @ Y3 @ C2 )
=> ( ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ X2
@ Y3 )
| ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ Y3
@ X2 ) ) ) ) ) ).
% subset.chain_total
thf(fact_1252_subset__Zorn,axiom,
! [A5: set_set_a] :
( ! [C6: set_set_a] :
( ( pred_chain_set_a @ A5 @ ord_less_set_a @ C6 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ C6 )
=> ( ord_less_eq_set_a @ Xa2 @ X4 ) ) ) )
=> ? [X: set_a] :
( ( member_set_a @ X @ A5 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A5 )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% subset_Zorn
thf(fact_1253_subset__chain__def,axiom,
! [A8: set_set_a,C7: set_set_a] :
( ( pred_chain_set_a @ A8 @ ord_less_set_a @ C7 )
= ( ( ord_le3724670747650509150_set_a @ C7 @ A8 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ C7 )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ C7 )
=> ( ( ord_less_eq_set_a @ X3 @ Y )
| ( ord_less_eq_set_a @ Y @ X3 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_1254_subset__chain__insert,axiom,
! [A8: set_set_a,B4: set_a,B9: set_set_a] :
( ( pred_chain_set_a @ A8 @ ord_less_set_a @ ( insert_set_a @ B4 @ B9 ) )
= ( ( member_set_a @ B4 @ A8 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ B9 )
=> ( ( ord_less_eq_set_a @ X3 @ B4 )
| ( ord_less_eq_set_a @ B4 @ X3 ) ) )
& ( pred_chain_set_a @ A8 @ ord_less_set_a @ B9 ) ) ) ).
% subset_chain_insert
thf(fact_1255_pred__on_Ochain__extend,axiom,
! [A5: set_set_a,P: set_a > set_a > $o,C2: set_set_a,Z: set_a] :
( ( pred_chain_set_a @ A5 @ P @ C2 )
=> ( ( member_set_a @ Z @ A5 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ C2 )
=> ( sup_su8131220024717662786et_a_o @ P
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ X
@ Z ) )
=> ( pred_chain_set_a @ A5 @ P @ ( sup_sup_set_set_a @ ( insert_set_a @ Z @ bot_bot_set_set_a ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1256_pred__on_Ochain__extend,axiom,
! [A5: set_a,P: a > a > $o,C2: set_a,Z: a] :
( ( pred_chain_a @ A5 @ P @ C2 )
=> ( ( member_a @ Z @ A5 )
=> ( ! [X: a] :
( ( member_a @ X @ C2 )
=> ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ X
@ Z ) )
=> ( pred_chain_a @ A5 @ P @ ( sup_sup_set_a @ ( insert_a @ Z @ bot_bot_set_a ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1257_reflclp__idemp,axiom,
! [P: a > a > $o] :
( ( sup_sup_a_a_o
@ ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) ) ) ).
% reflclp_idemp
thf(fact_1258_chain__subset__alt__def,axiom,
( chain_subset_a
= ( pred_chain_set_a @ top_top_set_set_a @ ord_less_set_a ) ) ).
% chain_subset_alt_def
thf(fact_1259_chain__subset__def,axiom,
( chain_subset_a
= ( ^ [C5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ C5 )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ C5 )
=> ( ( ord_less_eq_set_a @ X3 @ Y )
| ( ord_less_eq_set_a @ Y @ X3 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_1260_image__Fpow__mono,axiom,
! [F: a > a,A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A5 ) @ B4 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ ( finite_Fpow_a @ A5 ) ) @ ( finite_Fpow_a @ B4 ) ) ) ).
% image_Fpow_mono
thf(fact_1261_finite__has__minimal2,axiom,
! [A5: set_a,A: a] :
( ( finite_finite_a @ A5 )
=> ( ( member_a @ A @ A5 )
=> ? [X: a] :
( ( member_a @ X @ A5 )
& ( ord_less_eq_a @ X @ A )
& ! [Xa: a] :
( ( member_a @ Xa @ A5 )
=> ( ( ord_less_eq_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1262_finite__has__maximal2,axiom,
! [A5: set_a,A: a] :
( ( finite_finite_a @ A5 )
=> ( ( member_a @ A @ A5 )
=> ? [X: a] :
( ( member_a @ X @ A5 )
& ( ord_less_eq_a @ A @ X )
& ! [Xa: a] :
( ( member_a @ Xa @ A5 )
=> ( ( ord_less_eq_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1263_finite__has__maximal,axiom,
! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ? [X: a] :
( ( member_a @ X @ A5 )
& ! [Xa: a] :
( ( member_a @ Xa @ A5 )
=> ( ( ord_less_eq_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1264_finite__has__minimal,axiom,
! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ? [X: a] :
( ( member_a @ X @ A5 )
& ! [Xa: a] :
( ( member_a @ Xa @ A5 )
=> ( ( ord_less_eq_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1265_finite__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_1266_finite__ne__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ! [X: a] : ( P @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ! [X: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ~ ( member_a @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1267_infinite__finite__induct,axiom,
! [P: set_a > $o,A5: set_a] :
( ! [A9: set_a] :
( ~ ( finite_finite_a @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X @ F4 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1268_finite__subset__induct_H,axiom,
! [F3: set_a,A5: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A3 @ A5 )
=> ( ( ord_less_eq_set_a @ F4 @ A5 )
=> ( ~ ( member_a @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A3 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1269_finite__subset__induct,axiom,
! [F3: set_a,A5: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A3 @ A5 )
=> ( ~ ( member_a @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1270_finite__empty__induct,axiom,
! [A5: set_a,P: set_a > $o] :
( ( finite_finite_a @ A5 )
=> ( ( P @ A5 )
=> ( ! [A3: a,A9: set_a] :
( ( finite_finite_a @ A9 )
=> ( ( member_a @ A3 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus_set_a @ A9 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1271_finite__remove__induct,axiom,
! [B4: set_a,P: set_a > $o] :
( ( finite_finite_a @ B4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A9: set_a] :
( ( finite_finite_a @ A9 )
=> ( ( A9 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A9 @ B4 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A9 )
=> ( P @ ( minus_minus_set_a @ A9 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_1272_remove__induct,axiom,
! [P: set_a > $o,B4: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B4 )
=> ( P @ B4 ) )
=> ( ! [A9: set_a] :
( ( finite_finite_a @ A9 )
=> ( ( A9 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A9 @ B4 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A9 )
=> ( P @ ( minus_minus_set_a @ A9 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_1273_infinite__growing,axiom,
! [X6: set_a] :
( ( X6 != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ X6 )
=> ? [Xa: a] :
( ( member_a @ Xa @ X6 )
& ( ord_less_a @ X @ Xa ) ) )
=> ~ ( finite_finite_a @ X6 ) ) ) ).
% infinite_growing
thf(fact_1274_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > a] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,S4: set_a] :
( ( finite_finite_a @ S4 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S4 )
=> ( ord_less_eq_a @ ( F @ Y5 ) @ ( F @ X ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert_a @ X @ S4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1275_arg__min__least,axiom,
! [S2: set_a,Y3: a,F: a > a] :
( ( finite_finite_a @ S2 )
=> ( ( S2 != bot_bot_set_a )
=> ( ( member_a @ Y3 @ S2 )
=> ( ord_less_eq_a @ ( F @ ( lattic3288624042836100505on_a_a @ F @ S2 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
% Conjectures (2)
thf(conj_0,hypothesis,
! [M6: a] :
( ( member_a @ M6 @ s )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ s )
=> ( ord_less_eq_a @ M6 @ Y2 ) )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------