TPTP Problem File: SLH0558^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Median_Method/0000_Median/prob_00052_001923__14614652_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1392 ( 447 unt; 117 typ; 0 def)
% Number of atoms : 4163 (1010 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11506 ( 416 ~; 81 |; 277 &;8769 @)
% ( 0 <=>;1963 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 512 ( 512 >; 0 *; 0 +; 0 <<)
% Number of symbols : 113 ( 112 usr; 7 con; 0-4 aty)
% Number of variables : 3838 ( 230 ^;3487 !; 121 ?;3838 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:42:57.637
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
set_set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (112)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple9105089376463352645_set_a: set_set_set_a > set_set_a ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__a_J,type,
comple6135023378680113637_set_a: set_set_a > set_a ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
condit1487729544555641870_set_a: set_set_set_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_Elementary__Topology_Oclosure_001tf__a,type,
elementary_closure_a: set_a > set_a ).
thf(sy_c_Extended__Real__Limits_Oorder__class_Omono__set_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
extend2184654222208438451_set_a: set_set_set_a > $o ).
thf(sy_c_Extended__Real__Limits_Oorder__class_Omono__set_001t__Set__Oset_Itf__a_J,type,
extend347329919781519187_set_a: set_set_a > $o ).
thf(sy_c_Extended__Real__Limits_Oorder__class_Omono__set_001tf__a,type,
extend2808419353335425523_set_a: set_a > $o ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_Itf__a_J,type,
finite_Fpow_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
finite7209287970140883943_set_a: set_set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
uminus6103902357914783669_set_a: set_set_a > set_set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
inf_in4918978645039662700_set_a: set_set_set_set_a > set_set_set_set_a > set_set_set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
inf_in1205276777018777868_set_a: set_set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
semila844772492610406764_set_a: ( set_set_a > set_set_a > set_set_a ) > set_set_a > ( set_set_a > set_set_a > $o ) > ( set_set_a > set_set_a > $o ) > $o ).
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_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
sup_su6872963709084814930_set_a: set_set_set_set_a > set_set_set_set_a > set_set_set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
sup_su2076012971530813682_set_a: set_set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Set__Oset_Itf__a_J_001tf__a,type,
lattic2642421085827009593et_a_a: ( set_a > a ) > set_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_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_Itf__a_J,type,
lattic8209813465164889211_set_a: set_set_a > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
lattic338143333561554293_set_a: set_set_set_a > set_set_a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_Itf__a_J,type,
lattic2918178356826803221_set_a: set_set_a > set_a ).
thf(sy_c_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Median_Oup__ray_001tf__a,type,
up_ray_a: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo3380559777022489994_set_a: set_set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
ord_le6641866341576993578_set_a: set_set_set_set_a > set_set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le8783759336481291722_set_a: set_set_set_a > set_set_set_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_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
ord_le8049040685576063006_set_a: set_set_set_set_a > set_set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
order_7241106464633449719_set_a: ( set_set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
order_3565860530148683671_set_a: ( set_set_a > $o ) > set_set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
order_Greatest_a: ( a > $o ) > a ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
orderi5875812994216768367_set_a: ( set_set_a > set_set_a > $o ) > ( set_set_a > set_set_a > $o ) > set_set_a > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_Itf__a_J,type,
ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
collec7971272359403355785_set_a: ( set_set_set_a > $o ) > set_set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1042221919965026181_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_4955109552351689957_set_a: ( set_a > set_set_a ) > set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_a_set_set_a: ( a > set_set_a ) > set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
insert_set_set_a: set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__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__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_or3932701788497876179_set_a: set_set_set_a > set_set_set_a > set_set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
set_or6288561110385358355_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001tf__a,type,
set_or672772299803893939Most_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_or4667047852671975087_set_a: set_set_set_a > set_set_set_a > set_set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or7717641506629902671_set_a: set_set_a > set_set_a > set_set_set_a ).
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001tf__a,type,
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set_ord_atLeast_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_or3788817117504262629_set_a: set_set_set_a > set_set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001tf__a,type,
set_ord_atMost_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
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thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001tf__a,type,
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thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
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thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or5544419673604628330_set_a: set_set_a > set_set_a > set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_Itf__a_J,type,
set_or6017932776736107018_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001tf__a,type,
set_or5939364468397584554Than_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_Itf__a_J,type,
set_or460448635090783044_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001tf__a,type,
set_or8632414552788122084Than_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_Itf__a_J,type,
set_or5421148953861284865_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001tf__a,type,
set_ord_lessThan_a: a > set_a ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__a,type,
topolo8477419352202985285open_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oclosed_001tf__a,type,
topolo784654279908865136osed_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Ocompact_001tf__a,type,
topolo8439159285038550427pact_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected_001tf__a,type,
topolo2370605967727889109cted_a: set_a > $o ).
thf(sy_c_Zorn_Ochains_001t__Set__Oset_Itf__a_J,type,
chains_set_a: set_set_set_a > set_set_set_set_a ).
thf(sy_c_Zorn_Ochains_001tf__a,type,
chains_a: set_set_a > set_set_set_a ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
member_set_set_set_a: set_set_set_a > set_set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (1274)
thf(fact_0_assms,axiom,
up_ray_a @ i ).
% assms
thf(fact_1_c,axiom,
member_a @ x @ i ).
% c
thf(fact_2_calculation,axiom,
ord_less_eq_set_a @ i @ ( set_ord_atLeast_a @ x ) ).
% calculation
thf(fact_3_b,axiom,
~ ( topolo784654279908865136osed_a @ i ) ).
% b
thf(fact_4_False,axiom,
~ ? [Y: a] :
( ( ord_less_a @ Y @ x )
& ( member_a @ Y @ i ) ) ).
% False
thf(fact_5_atLeast__subset__iff,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le8049040685576063006_set_a @ ( set_or4597695895147092457_set_a @ X ) @ ( set_or4597695895147092457_set_a @ Y2 ) )
= ( ord_le5722252365846178494_set_a @ Y2 @ X ) ) ).
% atLeast_subset_iff
thf(fact_6_atLeast__subset__iff,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or3904034815786525833_set_a @ X ) @ ( set_or3904034815786525833_set_a @ Y2 ) )
= ( ord_le3724670747650509150_set_a @ Y2 @ X ) ) ).
% atLeast_subset_iff
thf(fact_7_atLeast__subset__iff,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or8362275514725411625_set_a @ X ) @ ( set_or8362275514725411625_set_a @ Y2 ) )
= ( ord_less_eq_set_a @ Y2 @ X ) ) ).
% atLeast_subset_iff
thf(fact_8_atLeast__subset__iff,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ X ) @ ( set_ord_atLeast_a @ Y2 ) )
= ( ord_less_eq_a @ Y2 @ X ) ) ).
% atLeast_subset_iff
thf(fact_9_atLeast__iff,axiom,
! [I: set_set_set_a,K: set_set_set_a] :
( ( member_set_set_set_a @ I @ ( set_or4597695895147092457_set_a @ K ) )
= ( ord_le5722252365846178494_set_a @ K @ I ) ) ).
% atLeast_iff
thf(fact_10_atLeast__iff,axiom,
! [I: set_set_a,K: set_set_a] :
( ( member_set_set_a @ I @ ( set_or3904034815786525833_set_a @ K ) )
= ( ord_le3724670747650509150_set_a @ K @ I ) ) ).
% atLeast_iff
thf(fact_11_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_12_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_13_atLeast__eq__iff,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ( set_or3904034815786525833_set_a @ X )
= ( set_or3904034815786525833_set_a @ Y2 ) )
= ( X = Y2 ) ) ).
% atLeast_eq_iff
thf(fact_14_atLeast__eq__iff,axiom,
! [X: set_a,Y2: set_a] :
( ( ( set_or8362275514725411625_set_a @ X )
= ( set_or8362275514725411625_set_a @ Y2 ) )
= ( X = Y2 ) ) ).
% atLeast_eq_iff
thf(fact_15_atLeast__eq__iff,axiom,
! [X: a,Y2: a] :
( ( ( set_ord_atLeast_a @ X )
= ( set_ord_atLeast_a @ Y2 ) )
= ( X = Y2 ) ) ).
% atLeast_eq_iff
thf(fact_16_subsetI,axiom,
! [A: set_set_set_set_a,B: set_set_set_set_a] :
( ! [X2: set_set_set_a] :
( ( member_set_set_set_a @ X2 @ A )
=> ( member_set_set_set_a @ X2 @ B ) )
=> ( ord_le8049040685576063006_set_a @ A @ B ) ) ).
% subsetI
thf(fact_17_subsetI,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( member_set_set_a @ X2 @ B ) )
=> ( ord_le5722252365846178494_set_a @ A @ B ) ) ).
% subsetI
thf(fact_18_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_19_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_20_subset__antisym,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( ord_le5722252365846178494_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_21_subset__antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_22_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_23_order__refl,axiom,
! [X: set_set_set_a] : ( ord_le5722252365846178494_set_a @ X @ X ) ).
% order_refl
thf(fact_24_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_25_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_26_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_27_dual__order_Orefl,axiom,
! [A2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_28_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_29_dual__order_Orefl,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_30_dual__order_Orefl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_31_up__ray__def,axiom,
( up_ray_a
= ( ^ [I2: set_a] :
! [X3: a,Y3: a] :
( ( member_a @ X3 @ I2 )
=> ( ( ord_less_eq_a @ X3 @ Y3 )
=> ( member_a @ Y3 @ I2 ) ) ) ) ) ).
% up_ray_def
thf(fact_32_interval__def,axiom,
( interval_a
= ( ^ [I2: set_a] :
! [X3: a,Y3: a,Z: a] :
( ( member_a @ X3 @ I2 )
=> ( ( member_a @ Z @ I2 )
=> ( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ Z )
=> ( member_a @ Y3 @ I2 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_33_in__mono,axiom,
! [A: set_set_set_set_a,B: set_set_set_set_a,X: set_set_set_a] :
( ( ord_le8049040685576063006_set_a @ A @ B )
=> ( ( member_set_set_set_a @ X @ A )
=> ( member_set_set_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_34_in__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a,X: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( member_set_set_a @ X @ A )
=> ( member_set_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_35_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_36_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_37_subsetD,axiom,
! [A: set_set_set_set_a,B: set_set_set_set_a,C: set_set_set_a] :
( ( ord_le8049040685576063006_set_a @ A @ B )
=> ( ( member_set_set_set_a @ C @ A )
=> ( member_set_set_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_38_subsetD,axiom,
! [A: set_set_set_a,B: set_set_set_a,C: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( member_set_set_a @ C @ A )
=> ( member_set_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_39_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_40_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_41_equalityE,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( A = B )
=> ~ ( ( ord_le5722252365846178494_set_a @ A @ B )
=> ~ ( ord_le5722252365846178494_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_42_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_43_equalityE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_44_less__imp__neq,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_less_set_set_a @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_45_less__imp__neq,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_46_less__imp__neq,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_47_less__imp__neq,axiom,
! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_48_order_Oasym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_set_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_49_order_Oasym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_50_order_Oasym,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A2 @ B2 )
=> ~ ( ord_le8783759336481291722_set_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_51_order_Oasym,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ~ ( ord_less_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_52_ord__eq__less__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A2 = B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_53_ord__eq__less__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_54_ord__eq__less__trans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_set_a] :
( ( A2 = B2 )
=> ( ( ord_le8783759336481291722_set_a @ B2 @ C )
=> ( ord_le8783759336481291722_set_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_55_ord__eq__less__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_56_ord__less__eq__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_57_ord__less__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_58_ord__less__eq__trans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le8783759336481291722_set_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_59_ord__less__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_60_antisym__conv3,axiom,
! [Y2: a,X: a] :
( ~ ( ord_less_a @ Y2 @ X )
=> ( ( ~ ( ord_less_a @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_61_linorder__cases,axiom,
! [X: a,Y2: a] :
( ~ ( ord_less_a @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_a @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_62_dual__order_Oasym,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_set_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_63_dual__order_Oasym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_64_dual__order_Oasym,axiom,
! [B2: set_set_set_a,A2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ B2 @ A2 )
=> ~ ( ord_le8783759336481291722_set_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_65_dual__order_Oasym,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ~ ( ord_less_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_66_dual__order_Oirrefl,axiom,
! [A2: set_set_a] :
~ ( ord_less_set_set_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_67_dual__order_Oirrefl,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_68_dual__order_Oirrefl,axiom,
! [A2: set_set_set_a] :
~ ( ord_le8783759336481291722_set_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_69_dual__order_Oirrefl,axiom,
! [A2: a] :
~ ( ord_less_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_70_linorder__less__wlog,axiom,
! [P: a > a > $o,A2: a,B2: a] :
( ! [A3: a,B3: a] :
( ( ord_less_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a] : ( P @ A3 @ A3 )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_71_order_Ostrict__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_72_order_Ostrict__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_73_order_Ostrict__trans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A2 @ B2 )
=> ( ( ord_le8783759336481291722_set_a @ B2 @ C )
=> ( ord_le8783759336481291722_set_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_74_order_Ostrict__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_75_not__less__iff__gr__or__eq,axiom,
! [X: a,Y2: a] :
( ( ~ ( ord_less_a @ X @ Y2 ) )
= ( ( ord_less_a @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_76_dual__order_Ostrict__trans,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ( ord_less_set_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_77_dual__order_Ostrict__trans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_78_dual__order_Ostrict__trans,axiom,
! [B2: set_set_set_a,A2: set_set_set_a,C: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ B2 @ A2 )
=> ( ( ord_le8783759336481291722_set_a @ C @ B2 )
=> ( ord_le8783759336481291722_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_79_dual__order_Ostrict__trans,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ( ord_less_a @ C @ B2 )
=> ( ord_less_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_80_order_Ostrict__implies__not__eq,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_81_order_Ostrict__implies__not__eq,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_82_order_Ostrict__implies__not__eq,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_83_order_Ostrict__implies__not__eq,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_84_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_85_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_86_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_set_set_a,A2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_87_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_88_linorder__neqE,axiom,
! [X: a,Y2: a] :
( ( X != Y2 )
=> ( ~ ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_89_order__less__asym,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_less_set_set_a @ X @ Y2 )
=> ~ ( ord_less_set_set_a @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_90_order__less__asym,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ~ ( ord_less_set_a @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_91_order__less__asym,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X @ Y2 )
=> ~ ( ord_le8783759336481291722_set_a @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_92_order__less__asym,axiom,
! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ~ ( ord_less_a @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_93_linorder__neq__iff,axiom,
! [X: a,Y2: a] :
( ( X != Y2 )
= ( ( ord_less_a @ X @ Y2 )
| ( ord_less_a @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_94_order__less__asym_H,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_set_a @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_95_order__less__asym_H,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_96_order__less__asym_H,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A2 @ B2 )
=> ~ ( ord_le8783759336481291722_set_a @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_97_order__less__asym_H,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ~ ( ord_less_a @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_98_order__less__trans,axiom,
! [X: set_set_a,Y2: set_set_a,Z2: set_set_a] :
( ( ord_less_set_set_a @ X @ Y2 )
=> ( ( ord_less_set_set_a @ Y2 @ Z2 )
=> ( ord_less_set_set_a @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_99_order__less__trans,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ( ord_less_set_a @ Y2 @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_100_order__less__trans,axiom,
! [X: set_set_set_a,Y2: set_set_set_a,Z2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X @ Y2 )
=> ( ( ord_le8783759336481291722_set_a @ Y2 @ Z2 )
=> ( ord_le8783759336481291722_set_a @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_101_order__less__trans,axiom,
! [X: a,Y2: a,Z2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ( ord_less_a @ Y2 @ Z2 )
=> ( ord_less_a @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_102_ord__eq__less__subst,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_103_ord__eq__less__subst,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_104_ord__eq__less__subst,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_105_ord__eq__less__subst,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_106_ord__eq__less__subst,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_107_ord__eq__less__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_108_ord__eq__less__subst,axiom,
! [A2: set_set_set_a,F: a > set_set_set_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_le8783759336481291722_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le8783759336481291722_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_109_ord__eq__less__subst,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_110_ord__eq__less__subst,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_111_ord__eq__less__subst,axiom,
! [A2: a,F: set_set_set_a > a,B2: set_set_set_a,C: set_set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le8783759336481291722_set_a @ B2 @ C )
=> ( ! [X2: set_set_set_a,Y4: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_112_ord__less__eq__subst,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_113_ord__less__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_114_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_115_ord__less__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_116_ord__less__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_117_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_118_ord__less__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_set_set_a,C: set_set_set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_le8783759336481291722_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le8783759336481291722_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_119_ord__less__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_120_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_121_ord__less__eq__subst,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,F: set_set_set_a > a,C: a] :
( ( ord_le8783759336481291722_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_set_set_a,Y4: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_122_order__less__irrefl,axiom,
! [X: set_set_a] :
~ ( ord_less_set_set_a @ X @ X ) ).
% order_less_irrefl
thf(fact_123_order__less__irrefl,axiom,
! [X: set_a] :
~ ( ord_less_set_a @ X @ X ) ).
% order_less_irrefl
thf(fact_124_order__less__irrefl,axiom,
! [X: set_set_set_a] :
~ ( ord_le8783759336481291722_set_a @ X @ X ) ).
% order_less_irrefl
thf(fact_125_order__less__irrefl,axiom,
! [X: a] :
~ ( ord_less_a @ X @ X ) ).
% order_less_irrefl
thf(fact_126_order__less__subst1,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_127_order__less__subst1,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_128_order__less__subst1,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_129_order__less__subst1,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_130_order__less__subst1,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( ord_less_set_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_131_order__less__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_132_order__less__subst1,axiom,
! [A2: a,F: set_set_set_a > a,B2: set_set_set_a,C: set_set_set_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le8783759336481291722_set_a @ B2 @ C )
=> ( ! [X2: set_set_set_a,Y4: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_133_order__less__subst1,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_134_order__less__subst1,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_135_order__less__subst1,axiom,
! [A2: set_set_set_a,F: a > set_set_set_a,B2: a,C: a] :
( ( ord_le8783759336481291722_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_le8783759336481291722_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le8783759336481291722_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_136_mem__Collect__eq,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( member_set_set_a @ A2 @ ( collect_set_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
! [A2: set_set_set_a,P: set_set_set_a > $o] :
( ( member_set_set_set_a @ A2 @ ( collec7971272359403355785_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_138_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_139_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A: set_set_set_a] :
( ( collect_set_set_a
@ ^ [X3: set_set_a] : ( member_set_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
! [A: set_set_set_set_a] :
( ( collec7971272359403355785_set_a
@ ^ [X3: set_set_set_a] : ( member_set_set_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_142_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_143_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_144_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_145_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_146_order__less__subst2,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_147_order__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_148_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_149_order__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_150_order__less__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_151_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_152_order__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_set_a,C: set_set_set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_le8783759336481291722_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_le8783759336481291722_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le8783759336481291722_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_153_order__less__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_154_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_155_order__less__subst2,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,F: set_set_set_a > a,C: a] :
( ( ord_le8783759336481291722_set_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_set_a,Y4: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_156_order__less__not__sym,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_less_set_set_a @ X @ Y2 )
=> ~ ( ord_less_set_set_a @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_157_order__less__not__sym,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ~ ( ord_less_set_a @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_158_order__less__not__sym,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X @ Y2 )
=> ~ ( ord_le8783759336481291722_set_a @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_159_order__less__not__sym,axiom,
! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ~ ( ord_less_a @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_160_order__less__imp__triv,axiom,
! [X: set_set_a,Y2: set_set_a,P: $o] :
( ( ord_less_set_set_a @ X @ Y2 )
=> ( ( ord_less_set_set_a @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_161_order__less__imp__triv,axiom,
! [X: set_a,Y2: set_a,P: $o] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ( ord_less_set_a @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_162_order__less__imp__triv,axiom,
! [X: set_set_set_a,Y2: set_set_set_a,P: $o] :
( ( ord_le8783759336481291722_set_a @ X @ Y2 )
=> ( ( ord_le8783759336481291722_set_a @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_163_order__less__imp__triv,axiom,
! [X: a,Y2: a,P: $o] :
( ( ord_less_a @ X @ Y2 )
=> ( ( ord_less_a @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_164_linorder__less__linear,axiom,
! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_a @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_165_order__less__imp__not__eq,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_less_set_set_a @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_166_order__less__imp__not__eq,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_167_order__less__imp__not__eq,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_168_order__less__imp__not__eq,axiom,
! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_169_order__less__imp__not__eq2,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_less_set_set_a @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_170_order__less__imp__not__eq2,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_171_order__less__imp__not__eq2,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_172_order__less__imp__not__eq2,axiom,
! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_173_order__less__imp__not__less,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_less_set_set_a @ X @ Y2 )
=> ~ ( ord_less_set_set_a @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_174_order__less__imp__not__less,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ~ ( ord_less_set_a @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_175_order__less__imp__not__less,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X @ Y2 )
=> ~ ( ord_le8783759336481291722_set_a @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_176_order__less__imp__not__less,axiom,
! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ~ ( ord_less_a @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_177_order__le__imp__less__or__eq,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ( ord_le8783759336481291722_set_a @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_178_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_set_a @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_179_order__le__imp__less__or__eq,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ( ord_less_a @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_180_order__le__imp__less__or__eq,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( ord_less_set_set_a @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_181_linorder__le__less__linear,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
| ( ord_less_a @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_182_order__less__le__subst2,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_183_order__less__le__subst2,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_184_order__less__le__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_185_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_186_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_187_order__less__le__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_188_order__less__le__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_189_order__less__le__subst2,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,F: set_set_set_a > a,C: a] :
( ( ord_le8783759336481291722_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_set_a,Y4: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_190_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_191_order__less__le__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_set_a,C: set_set_set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_le8783759336481291722_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le8783759336481291722_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_192_order__less__le__subst1,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_193_order__less__le__subst1,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_194_order__less__le__subst1,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_195_order__less__le__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_196_order__less__le__subst1,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( ord_less_set_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_197_order__less__le__subst1,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_198_order__less__le__subst1,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_199_order__less__le__subst1,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_200_order__less__le__subst1,axiom,
! [A2: set_set_a,F: set_set_a > set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_201_order__less__le__subst1,axiom,
! [A2: set_set_set_a,F: a > set_set_set_a,B2: a,C: a] :
( ( ord_le8783759336481291722_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le8783759336481291722_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_202_order__le__less__subst2,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_203_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_204_order__le__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_205_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_206_order__le__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_207_order__le__less__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_208_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_209_order__le__less__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_210_order__le__less__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_211_order__le__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_set_a,C: set_set_set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_le8783759336481291722_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le8783759336481291722_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_212_order__le__less__subst1,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_213_order__le__less__subst1,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_214_order__le__less__subst1,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_215_order__le__less__subst1,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_216_order__le__less__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_217_order__le__less__subst1,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_218_order__le__less__subst1,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y4 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_219_order__le__less__subst1,axiom,
! [A2: a,F: set_set_set_a > a,B2: set_set_set_a,C: set_set_set_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le8783759336481291722_set_a @ B2 @ C )
=> ( ! [X2: set_set_set_a,Y4: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X2 @ Y4 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_220_order__le__less__subst1,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_set_a @ X2 @ Y4 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_221_order__le__less__subst1,axiom,
! [A2: set_set_set_a,F: a > set_set_set_a,B2: a,C: a] :
( ( ord_le5722252365846178494_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_a @ X2 @ Y4 )
=> ( ord_le8783759336481291722_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le8783759336481291722_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_222_order__less__le__trans,axiom,
! [X: set_set_set_a,Y2: set_set_set_a,Z2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X @ Y2 )
=> ( ( ord_le5722252365846178494_set_a @ Y2 @ Z2 )
=> ( ord_le8783759336481291722_set_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_223_order__less__le__trans,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_224_order__less__le__trans,axiom,
! [X: a,Y2: a,Z2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ( ord_less_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_225_order__less__le__trans,axiom,
! [X: set_set_a,Y2: set_set_a,Z2: set_set_a] :
( ( ord_less_set_set_a @ X @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ Z2 )
=> ( ord_less_set_set_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_226_order__le__less__trans,axiom,
! [X: set_set_set_a,Y2: set_set_set_a,Z2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ( ord_le8783759336481291722_set_a @ Y2 @ Z2 )
=> ( ord_le8783759336481291722_set_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_227_order__le__less__trans,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_set_a @ Y2 @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_228_order__le__less__trans,axiom,
! [X: a,Y2: a,Z2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ( ord_less_a @ Y2 @ Z2 )
=> ( ord_less_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_229_order__le__less__trans,axiom,
! [X: set_set_a,Y2: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( ord_less_set_set_a @ Y2 @ Z2 )
=> ( ord_less_set_set_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_230_order__neq__le__trans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( A2 != B2 )
=> ( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ord_le8783759336481291722_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_231_order__neq__le__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_232_order__neq__le__trans,axiom,
! [A2: a,B2: a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_a @ A2 @ B2 )
=> ( ord_less_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_233_order__neq__le__trans,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 != B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_234_order__le__neq__trans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le8783759336481291722_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_235_order__le__neq__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_236_order__le__neq__trans,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_237_order__le__neq__trans,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_238_order__less__imp__le,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_239_order__less__imp__le,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_240_order__less__imp__le,axiom,
! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_eq_a @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_241_order__less__imp__le,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_less_set_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_242_linorder__not__less,axiom,
! [X: a,Y2: a] :
( ( ~ ( ord_less_a @ X @ Y2 ) )
= ( ord_less_eq_a @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_243_linorder__not__le,axiom,
! [X: a,Y2: a] :
( ( ~ ( ord_less_eq_a @ X @ Y2 ) )
= ( ord_less_a @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_244_order__less__le,axiom,
( ord_le8783759336481291722_set_a
= ( ^ [X3: set_set_set_a,Y3: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_245_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_246_order__less__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_247_order__less__le,axiom,
( ord_less_set_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_248_order__le__less,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [X3: set_set_set_a,Y3: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_249_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_250_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X3: a,Y3: a] :
( ( ord_less_a @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_251_order__le__less,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_252_dual__order_Ostrict__implies__order,axiom,
! [B2: set_set_set_a,A2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ B2 @ A2 )
=> ( ord_le5722252365846178494_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_253_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_254_dual__order_Ostrict__implies__order,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ord_less_eq_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_255_dual__order_Ostrict__implies__order,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_256_order_Ostrict__implies__order,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A2 @ B2 )
=> ( ord_le5722252365846178494_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_257_order_Ostrict__implies__order,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_258_order_Ostrict__implies__order,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ord_less_eq_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_259_order_Ostrict__implies__order,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_260_dual__order_Ostrict__iff__not,axiom,
( ord_le8783759336481291722_set_a
= ( ^ [B4: set_set_set_a,A4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B4 @ A4 )
& ~ ( ord_le5722252365846178494_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_261_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_262_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ~ ( ord_less_eq_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_263_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ~ ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_264_dual__order_Ostrict__trans2,axiom,
! [B2: set_set_set_a,A2: set_set_set_a,C: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ B2 @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ C @ B2 )
=> ( ord_le8783759336481291722_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_265_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_266_dual__order_Ostrict__trans2,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ C @ B2 )
=> ( ord_less_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_267_dual__order_Ostrict__trans2,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_268_dual__order_Ostrict__trans1,axiom,
! [B2: set_set_set_a,A2: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ A2 )
=> ( ( ord_le8783759336481291722_set_a @ C @ B2 )
=> ( ord_le8783759336481291722_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_269_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_270_dual__order_Ostrict__trans1,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_a @ C @ B2 )
=> ( ord_less_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_271_dual__order_Ostrict__trans1,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_less_set_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_272_dual__order_Ostrict__iff__order,axiom,
( ord_le8783759336481291722_set_a
= ( ^ [B4: set_set_set_a,A4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_273_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_274_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_275_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_276_dual__order_Oorder__iff__strict,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [B4: set_set_set_a,A4: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_277_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_278_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_279_dual__order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_less_set_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_280_order_Ostrict__iff__not,axiom,
( ord_le8783759336481291722_set_a
= ( ^ [A4: set_set_set_a,B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A4 @ B4 )
& ~ ( ord_le5722252365846178494_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_281_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_282_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ~ ( ord_less_eq_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_283_order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ~ ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_284_order_Ostrict__trans2,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A2 @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ C )
=> ( ord_le8783759336481291722_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_285_order_Ostrict__trans2,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_286_order_Ostrict__trans2,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_287_order_Ostrict__trans2,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_288_order_Ostrict__trans1,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( ord_le8783759336481291722_set_a @ B2 @ C )
=> ( ord_le8783759336481291722_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_289_order_Ostrict__trans1,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_290_order_Ostrict__trans1,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_291_order_Ostrict__trans1,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_292_order_Ostrict__iff__order,axiom,
( ord_le8783759336481291722_set_a
= ( ^ [A4: set_set_set_a,B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_293_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_294_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_295_order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_296_order_Oorder__iff__strict,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A4: set_set_set_a,B4: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_297_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_298_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_299_order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_less_set_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_300_not__le__imp__less,axiom,
! [Y2: a,X: a] :
( ~ ( ord_less_eq_a @ Y2 @ X )
=> ( ord_less_a @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_301_less__le__not__le,axiom,
( ord_le8783759336481291722_set_a
= ( ^ [X3: set_set_set_a,Y3: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X3 @ Y3 )
& ~ ( ord_le5722252365846178494_set_a @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_302_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
& ~ ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_303_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
& ~ ( ord_less_eq_a @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_304_less__le__not__le,axiom,
( ord_less_set_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
& ~ ( ord_le3724670747650509150_set_a @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_305_antisym__conv2,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ( ~ ( ord_le8783759336481291722_set_a @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_306_antisym__conv2,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ~ ( ord_less_set_a @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_307_antisym__conv2,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ( ~ ( ord_less_a @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_308_antisym__conv2,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( ~ ( ord_less_set_set_a @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_309_antisym__conv1,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ~ ( ord_le8783759336481291722_set_a @ X @ Y2 )
=> ( ( ord_le5722252365846178494_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_310_antisym__conv1,axiom,
! [X: set_a,Y2: set_a] :
( ~ ( ord_less_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_311_antisym__conv1,axiom,
! [X: a,Y2: a] :
( ~ ( ord_less_a @ X @ Y2 )
=> ( ( ord_less_eq_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_312_antisym__conv1,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ~ ( ord_less_set_set_a @ X @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_313_nless__le,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ~ ( ord_le8783759336481291722_set_a @ A2 @ B2 ) )
= ( ~ ( ord_le5722252365846178494_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_314_nless__le,axiom,
! [A2: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_315_nless__le,axiom,
! [A2: a,B2: a] :
( ( ~ ( ord_less_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_316_nless__le,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ~ ( ord_less_set_set_a @ A2 @ B2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_317_leI,axiom,
! [X: a,Y2: a] :
( ~ ( ord_less_a @ X @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X ) ) ).
% leI
thf(fact_318_leD,axiom,
! [Y2: set_set_set_a,X: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ Y2 @ X )
=> ~ ( ord_le8783759336481291722_set_a @ X @ Y2 ) ) ).
% leD
thf(fact_319_leD,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ~ ( ord_less_set_a @ X @ Y2 ) ) ).
% leD
thf(fact_320_leD,axiom,
! [Y2: a,X: a] :
( ( ord_less_eq_a @ Y2 @ X )
=> ~ ( ord_less_a @ X @ Y2 ) ) ).
% leD
thf(fact_321_leD,axiom,
! [Y2: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X )
=> ~ ( ord_less_set_set_a @ X @ Y2 ) ) ).
% leD
thf(fact_322_order__antisym__conv,axiom,
! [Y2: set_set_set_a,X: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ Y2 @ X )
=> ( ( ord_le5722252365846178494_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_323_order__antisym__conv,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_324_order__antisym__conv,axiom,
! [Y2: a,X: a] :
( ( ord_less_eq_a @ Y2 @ X )
=> ( ( ord_less_eq_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_325_order__antisym__conv,axiom,
! [Y2: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_326_linorder__le__cases,axiom,
! [X: a,Y2: a] :
( ~ ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_327_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_328_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_329_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_330_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_331_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_332_ord__le__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_333_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_334_ord__le__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_335_ord__le__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_336_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_set_set_a,C: set_set_set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le5722252365846178494_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_337_ord__eq__le__subst,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_338_ord__eq__le__subst,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_339_ord__eq__le__subst,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_340_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_341_ord__eq__le__subst,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_342_ord__eq__le__subst,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_343_ord__eq__le__subst,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_344_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_345_ord__eq__le__subst,axiom,
! [A2: set_set_a,F: set_set_a > set_set_a,B2: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_346_ord__eq__le__subst,axiom,
! [A2: set_set_set_a,F: a > set_set_set_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le5722252365846178494_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_347_linorder__linear,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
| ( ord_less_eq_a @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_348_order__eq__refl,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( X = Y2 )
=> ( ord_le5722252365846178494_set_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_349_order__eq__refl,axiom,
! [X: set_a,Y2: set_a] :
( ( X = Y2 )
=> ( ord_less_eq_set_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_350_order__eq__refl,axiom,
! [X: a,Y2: a] :
( ( X = Y2 )
=> ( ord_less_eq_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_351_order__eq__refl,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( X = Y2 )
=> ( ord_le3724670747650509150_set_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_352_order__subst2,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_353_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_354_order__subst2,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_355_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_356_order__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_357_order__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > a,C: a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_358_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_359_order__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_360_order__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_361_order__subst2,axiom,
! [A2: a,B2: a,F: a > set_set_set_a,C: set_set_set_a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le5722252365846178494_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_362_order__subst1,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_363_order__subst1,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_364_order__subst1,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_365_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_366_order__subst1,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_367_order__subst1,axiom,
! [A2: set_set_a,F: a > set_set_a,B2: a,C: a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y4: a] :
( ( ord_less_eq_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_368_order__subst1,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_369_order__subst1,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_370_order__subst1,axiom,
! [A2: set_set_a,F: set_set_a > set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_371_order__subst1,axiom,
! [A2: a,F: set_set_set_a > a,B2: set_set_set_a,C: set_set_set_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ C )
=> ( ! [X2: set_set_set_a,Y4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X2 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_372_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_set_a,Z3: set_set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_set_set_a,B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A4 @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_373_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_374_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: a,Z3: a] : ( Y5 = Z3 ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ( ord_less_eq_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_375_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_376_antisym,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_377_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_378_antisym,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_379_antisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_380_dual__order_Otrans,axiom,
! [B2: set_set_set_a,A2: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ C @ B2 )
=> ( ord_le5722252365846178494_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_381_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_382_dual__order_Otrans,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ C @ B2 )
=> ( ord_less_eq_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_383_dual__order_Otrans,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_384_dual__order_Oantisym,axiom,
! [B2: set_set_set_a,A2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_385_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_386_dual__order_Oantisym,axiom,
! [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_387_dual__order_Oantisym,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_388_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_set_a,Z3: set_set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_set_set_a,B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B4 @ A4 )
& ( ord_le5722252365846178494_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_389_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_390_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: a,Z3: a] : ( Y5 = Z3 ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ( ord_less_eq_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_391_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_392_linorder__wlog,axiom,
! [P: a > a > $o,A2: a,B2: a] :
( ! [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_393_order__trans,axiom,
! [X: set_set_set_a,Y2: set_set_set_a,Z2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ( ord_le5722252365846178494_set_a @ Y2 @ Z2 )
=> ( ord_le5722252365846178494_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_394_order__trans,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_395_order__trans,axiom,
! [X: a,Y2: a,Z2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ( ord_less_eq_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_396_order__trans,axiom,
! [X: set_set_a,Y2: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_397_order_Otrans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ C )
=> ( ord_le5722252365846178494_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_398_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_399_order_Otrans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_400_order_Otrans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_401_order__antisym,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ( ord_le5722252365846178494_set_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_402_order__antisym,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_403_order__antisym,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_404_order__antisym,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_405_ord__le__eq__trans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le5722252365846178494_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_406_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_407_ord__le__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_408_ord__le__eq__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_409_ord__eq__le__trans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_set_a] :
( ( A2 = B2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ C )
=> ( ord_le5722252365846178494_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_410_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_411_ord__eq__le__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_412_ord__eq__le__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A2 = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_413_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_set_a,Z3: set_set_set_a] : ( Y5 = Z3 ) )
= ( ^ [X3: set_set_set_a,Y3: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X3 @ Y3 )
& ( ord_le5722252365846178494_set_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_414_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_415_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: a,Z3: a] : ( Y5 = Z3 ) )
= ( ^ [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
& ( ord_less_eq_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_416_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
& ( ord_le3724670747650509150_set_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_417_le__cases3,axiom,
! [X: a,Y2: a,Z2: a] :
( ( ( ord_less_eq_a @ X @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_a @ Y2 @ X )
=> ~ ( ord_less_eq_a @ X @ Z2 ) )
=> ( ( ( ord_less_eq_a @ X @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_a @ Z2 @ Y2 )
=> ~ ( ord_less_eq_a @ Y2 @ X ) )
=> ( ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_a @ Z2 @ X )
=> ~ ( ord_less_eq_a @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_418_nle__le,axiom,
! [A2: a,B2: a] :
( ( ~ ( ord_less_eq_a @ A2 @ B2 ) )
= ( ( ord_less_eq_a @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_419_Collect__mono__iff,axiom,
! [P: set_set_a > $o,Q: set_set_a > $o] :
( ( ord_le5722252365846178494_set_a @ ( collect_set_set_a @ P ) @ ( collect_set_set_a @ Q ) )
= ( ! [X3: set_set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_420_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_421_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_422_set__eq__subset,axiom,
( ( ^ [Y5: set_set_set_a,Z3: set_set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A5: set_set_set_a,B5: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A5 @ B5 )
& ( ord_le5722252365846178494_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_423_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_424_set__eq__subset,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_425_subset__trans,axiom,
! [A: set_set_set_a,B: set_set_set_a,C2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( ord_le5722252365846178494_set_a @ B @ C2 )
=> ( ord_le5722252365846178494_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_426_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_427_subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_428_Collect__mono,axiom,
! [P: set_set_a > $o,Q: set_set_a > $o] :
( ! [X2: set_set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le5722252365846178494_set_a @ ( collect_set_set_a @ P ) @ ( collect_set_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_429_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_430_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_431_subset__refl,axiom,
! [A: set_set_set_a] : ( ord_le5722252365846178494_set_a @ A @ A ) ).
% subset_refl
thf(fact_432_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_433_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_434_subset__iff,axiom,
( ord_le8049040685576063006_set_a
= ( ^ [A5: set_set_set_set_a,B5: set_set_set_set_a] :
! [T: set_set_set_a] :
( ( member_set_set_set_a @ T @ A5 )
=> ( member_set_set_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_435_subset__iff,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A5: set_set_set_a,B5: set_set_set_a] :
! [T: set_set_a] :
( ( member_set_set_a @ T @ A5 )
=> ( member_set_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_436_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_437_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A5 )
=> ( member_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_438_equalityD2,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( A = B )
=> ( ord_le5722252365846178494_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_439_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_440_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_441_equalityD1,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( A = B )
=> ( ord_le5722252365846178494_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_442_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_443_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_444_subset__eq,axiom,
( ord_le8049040685576063006_set_a
= ( ^ [A5: set_set_set_set_a,B5: set_set_set_set_a] :
! [X3: set_set_set_a] :
( ( member_set_set_set_a @ X3 @ A5 )
=> ( member_set_set_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_445_subset__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A5: set_set_set_a,B5: set_set_set_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A5 )
=> ( member_set_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_446_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_447_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( member_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_448_closed__atLeast,axiom,
! [A2: a] : ( topolo784654279908865136osed_a @ ( set_ord_atLeast_a @ A2 ) ) ).
% closed_atLeast
thf(fact_449_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_450_pinf_I6_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ~ ( ord_less_eq_a @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_451_pinf_I8_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( ord_less_eq_a @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_452_minf_I6_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( ord_less_eq_a @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_453_minf_I8_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ~ ( ord_less_eq_a @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_454_Greatest__equality,axiom,
! [P: set_set_set_a > $o,X: set_set_set_a] :
( ( P @ X )
=> ( ! [Y4: set_set_set_a] :
( ( P @ Y4 )
=> ( ord_le5722252365846178494_set_a @ Y4 @ X ) )
=> ( ( order_7241106464633449719_set_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_455_Greatest__equality,axiom,
! [P: set_a > $o,X: set_a] :
( ( P @ X )
=> ( ! [Y4: set_a] :
( ( P @ Y4 )
=> ( ord_less_eq_set_a @ Y4 @ X ) )
=> ( ( order_Greatest_set_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_456_Greatest__equality,axiom,
! [P: a > $o,X: a] :
( ( P @ X )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X ) )
=> ( ( order_Greatest_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_457_Greatest__equality,axiom,
! [P: set_set_a > $o,X: set_set_a] :
( ( P @ X )
=> ( ! [Y4: set_set_a] :
( ( P @ Y4 )
=> ( ord_le3724670747650509150_set_a @ Y4 @ X ) )
=> ( ( order_3565860530148683671_set_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_458_GreatestI2__order,axiom,
! [P: set_set_set_a > $o,X: set_set_set_a,Q: set_set_set_a > $o] :
( ( P @ X )
=> ( ! [Y4: set_set_set_a] :
( ( P @ Y4 )
=> ( ord_le5722252365846178494_set_a @ Y4 @ X ) )
=> ( ! [X2: set_set_set_a] :
( ( P @ X2 )
=> ( ! [Y: set_set_set_a] :
( ( P @ Y )
=> ( ord_le5722252365846178494_set_a @ Y @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_7241106464633449719_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_459_GreatestI2__order,axiom,
! [P: set_a > $o,X: set_a,Q: set_a > $o] :
( ( P @ X )
=> ( ! [Y4: set_a] :
( ( P @ Y4 )
=> ( ord_less_eq_set_a @ Y4 @ X ) )
=> ( ! [X2: set_a] :
( ( P @ X2 )
=> ( ! [Y: set_a] :
( ( P @ Y )
=> ( ord_less_eq_set_a @ Y @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_460_GreatestI2__order,axiom,
! [P: a > $o,X: a,Q: a > $o] :
( ( P @ X )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X ) )
=> ( ! [X2: a] :
( ( P @ X2 )
=> ( ! [Y: a] :
( ( P @ Y )
=> ( ord_less_eq_a @ Y @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_461_GreatestI2__order,axiom,
! [P: set_set_a > $o,X: set_set_a,Q: set_set_a > $o] :
( ( P @ X )
=> ( ! [Y4: set_set_a] :
( ( P @ Y4 )
=> ( ord_le3724670747650509150_set_a @ Y4 @ X ) )
=> ( ! [X2: set_set_a] :
( ( P @ X2 )
=> ( ! [Y: set_set_a] :
( ( P @ Y )
=> ( ord_le3724670747650509150_set_a @ Y @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_3565860530148683671_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_462_Icc__subset__Ici__iff,axiom,
! [L: set_set_set_a,H: set_set_set_a,L2: set_set_set_a] :
( ( ord_le8049040685576063006_set_a @ ( set_or3932701788497876179_set_a @ L @ H ) @ ( set_or4597695895147092457_set_a @ L2 ) )
= ( ~ ( ord_le5722252365846178494_set_a @ L @ H )
| ( ord_le5722252365846178494_set_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_463_Icc__subset__Ici__iff,axiom,
! [L: set_set_a,H: set_set_a,L2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ L @ H ) @ ( set_or3904034815786525833_set_a @ L2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ L @ H )
| ( ord_le3724670747650509150_set_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_464_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_465_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_466_psubsetI,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_le8783759336481291722_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_467_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_468_psubsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_469_Icc__eq__Icc,axiom,
! [L: set_set_set_a,H: set_set_set_a,L2: set_set_set_a,H2: set_set_set_a] :
( ( ( set_or3932701788497876179_set_a @ L @ H )
= ( set_or3932701788497876179_set_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le5722252365846178494_set_a @ L @ H )
& ~ ( ord_le5722252365846178494_set_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_470_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_471_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_472_Icc__eq__Icc,axiom,
! [L: set_set_a,H: set_set_a,L2: set_set_a,H2: set_set_a] :
( ( ( set_or4761464488706262899_set_a @ L @ H )
= ( set_or4761464488706262899_set_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le3724670747650509150_set_a @ L @ H )
& ~ ( ord_le3724670747650509150_set_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_473_atLeastAtMost__iff,axiom,
! [I: set_set_set_a,L: set_set_set_a,U: set_set_set_a] :
( ( member_set_set_set_a @ I @ ( set_or3932701788497876179_set_a @ L @ U ) )
= ( ( ord_le5722252365846178494_set_a @ L @ I )
& ( ord_le5722252365846178494_set_a @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_474_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_475_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_476_atLeastAtMost__iff,axiom,
! [I: set_set_a,L: set_set_a,U: set_set_a] :
( ( member_set_set_a @ I @ ( set_or4761464488706262899_set_a @ L @ U ) )
= ( ( ord_le3724670747650509150_set_a @ L @ I )
& ( ord_le3724670747650509150_set_a @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_477_atLeastatMost__subset__iff,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_set_a,D: set_set_set_a] :
( ( ord_le8049040685576063006_set_a @ ( set_or3932701788497876179_set_a @ A2 @ B2 ) @ ( set_or3932701788497876179_set_a @ C @ D ) )
= ( ~ ( ord_le5722252365846178494_set_a @ A2 @ B2 )
| ( ( ord_le5722252365846178494_set_a @ C @ A2 )
& ( ord_le5722252365846178494_set_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_478_atLeastatMost__subset__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a,D: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ A2 @ B2 ) @ ( set_or4761464488706262899_set_a @ C @ D ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 )
| ( ( ord_le3724670747650509150_set_a @ C @ A2 )
& ( ord_le3724670747650509150_set_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_479_atLeastatMost__subset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_480_atLeastatMost__subset__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_481_closed__atLeastAtMost,axiom,
! [A2: a,B2: a] : ( topolo784654279908865136osed_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) ) ).
% closed_atLeastAtMost
thf(fact_482_psubsetE,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A @ B )
=> ~ ( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ord_le5722252365846178494_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_483_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_484_psubsetE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_485_psubset__eq,axiom,
( ord_le8783759336481291722_set_a
= ( ^ [A5: set_set_set_a,B5: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_486_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_487_psubset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_488_psubset__imp__subset,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A @ B )
=> ( ord_le5722252365846178494_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_489_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_490_psubset__imp__subset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_491_psubset__subset__trans,axiom,
! [A: set_set_set_a,B: set_set_set_a,C2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A @ B )
=> ( ( ord_le5722252365846178494_set_a @ B @ C2 )
=> ( ord_le8783759336481291722_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_492_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_493_psubset__subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_less_set_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_494_subset__not__subset__eq,axiom,
( ord_le8783759336481291722_set_a
= ( ^ [A5: set_set_set_a,B5: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A5 @ B5 )
& ~ ( ord_le5722252365846178494_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_495_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_496_subset__not__subset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ~ ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_497_subset__psubset__trans,axiom,
! [A: set_set_set_a,B: set_set_set_a,C2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( ord_le8783759336481291722_set_a @ B @ C2 )
=> ( ord_le8783759336481291722_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_498_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_499_subset__psubset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_set_set_a @ B @ C2 )
=> ( ord_less_set_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_500_subset__iff__psubset__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A5: set_set_set_a,B5: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_501_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_502_subset__iff__psubset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_less_set_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_503_atLeastatMost__psubset__iff,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_set_a,D: set_set_set_a] :
( ( ord_le6641866341576993578_set_a @ ( set_or3932701788497876179_set_a @ A2 @ B2 ) @ ( set_or3932701788497876179_set_a @ C @ D ) )
= ( ( ~ ( ord_le5722252365846178494_set_a @ A2 @ B2 )
| ( ( ord_le5722252365846178494_set_a @ C @ A2 )
& ( ord_le5722252365846178494_set_a @ B2 @ D )
& ( ( ord_le8783759336481291722_set_a @ C @ A2 )
| ( ord_le8783759336481291722_set_a @ B2 @ D ) ) ) )
& ( ord_le5722252365846178494_set_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_504_atLeastatMost__psubset__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ord_less_set_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ B2 @ D )
& ( ( ord_less_set_a @ C @ A2 )
| ( ord_less_set_a @ B2 @ D ) ) ) )
& ( ord_less_eq_set_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_505_atLeastatMost__psubset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D )
& ( ( ord_less_a @ C @ A2 )
| ( ord_less_a @ B2 @ D ) ) ) )
& ( ord_less_eq_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_506_atLeastatMost__psubset__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a,D: set_set_a] :
( ( ord_le8783759336481291722_set_a @ ( set_or4761464488706262899_set_a @ A2 @ B2 ) @ ( set_or4761464488706262899_set_a @ C @ D ) )
= ( ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 )
| ( ( ord_le3724670747650509150_set_a @ C @ A2 )
& ( ord_le3724670747650509150_set_a @ B2 @ D )
& ( ( ord_less_set_set_a @ C @ A2 )
| ( ord_less_set_set_a @ B2 @ D ) ) ) )
& ( ord_le3724670747650509150_set_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_507_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_508_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_509_verit__comp__simplify1_I2_J,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_510_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_511_verit__la__disequality,axiom,
! [A2: a,B2: a] :
( ( A2 = B2 )
| ~ ( ord_less_eq_a @ A2 @ B2 )
| ~ ( ord_less_eq_a @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_512_verit__comp__simplify1_I1_J,axiom,
! [A2: set_set_a] :
~ ( ord_less_set_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_513_verit__comp__simplify1_I1_J,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_514_verit__comp__simplify1_I1_J,axiom,
! [A2: set_set_set_a] :
~ ( ord_le8783759336481291722_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_515_verit__comp__simplify1_I1_J,axiom,
! [A2: a] :
~ ( ord_less_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_516_pinf_I1_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X2: a] :
( ( ord_less_a @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z5: a] :
! [X2: a] :
( ( ord_less_a @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_517_pinf_I2_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X2: a] :
( ( ord_less_a @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z5: a] :
! [X2: a] :
( ( ord_less_a @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_518_pinf_I3_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_519_pinf_I4_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_520_pinf_I5_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ~ ( ord_less_a @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_521_pinf_I7_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ Z4 @ X4 )
=> ( ord_less_a @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_522_minf_I1_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z5: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_523_minf_I2_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z5: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z5: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_524_minf_I3_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_525_minf_I4_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_526_minf_I5_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ( ord_less_a @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_527_minf_I7_J,axiom,
! [T2: a] :
? [Z4: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z4 )
=> ~ ( ord_less_a @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_528_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_set_a,D: set_set_set_a] :
( ( ord_le8049040685576063006_set_a @ ( set_or3932701788497876179_set_a @ A2 @ B2 ) @ ( set_or4667047852671975087_set_a @ C @ D ) )
= ( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ C @ A2 )
& ( ord_le8783759336481291722_set_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_529_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a,D: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ A2 @ B2 ) @ ( set_or7717641506629902671_set_a @ C @ D ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ A2 )
& ( ord_less_set_set_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_530_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or5139330845457685135Than_a @ C @ D ) )
= ( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_531_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or2348907005316661231_set_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_set_a @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_532_Icc__subset__Iic__iff,axiom,
! [L: set_set_set_a,H: set_set_set_a,H2: set_set_set_a] :
( ( ord_le8049040685576063006_set_a @ ( set_or3932701788497876179_set_a @ L @ H ) @ ( set_or3788817117504262629_set_a @ H2 ) )
= ( ~ ( ord_le5722252365846178494_set_a @ L @ H )
| ( ord_le5722252365846178494_set_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_533_Icc__subset__Iic__iff,axiom,
! [L: set_set_a,H: set_set_a,H2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4761464488706262899_set_a @ L @ H ) @ ( set_or4016371710855203973_set_a @ H2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ L @ H )
| ( ord_le3724670747650509150_set_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_534_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_535_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_536_atMost__eq__iff,axiom,
! [X: set_a,Y2: set_a] :
( ( ( set_ord_atMost_set_a @ X )
= ( set_ord_atMost_set_a @ Y2 ) )
= ( X = Y2 ) ) ).
% atMost_eq_iff
thf(fact_537_atMost__eq__iff,axiom,
! [X: a,Y2: a] :
( ( ( set_ord_atMost_a @ X )
= ( set_ord_atMost_a @ Y2 ) )
= ( X = Y2 ) ) ).
% atMost_eq_iff
thf(fact_538_atMost__eq__iff,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ( set_or4016371710855203973_set_a @ X )
= ( set_or4016371710855203973_set_a @ Y2 ) )
= ( X = Y2 ) ) ).
% atMost_eq_iff
thf(fact_539_atMost__iff,axiom,
! [I: set_set_set_a,K: set_set_set_a] :
( ( member_set_set_set_a @ I @ ( set_or3788817117504262629_set_a @ K ) )
= ( ord_le5722252365846178494_set_a @ I @ K ) ) ).
% atMost_iff
thf(fact_540_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_541_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_542_atMost__iff,axiom,
! [I: set_set_a,K: set_set_a] :
( ( member_set_set_a @ I @ ( set_or4016371710855203973_set_a @ K ) )
= ( ord_le3724670747650509150_set_a @ I @ K ) ) ).
% atMost_iff
thf(fact_543_greaterThanLessThan__iff,axiom,
! [I: set_set_set_a,L: set_set_set_a,U: set_set_set_a] :
( ( member_set_set_set_a @ I @ ( set_or5684312773313237194_set_a @ L @ U ) )
= ( ( ord_le8783759336481291722_set_a @ L @ I )
& ( ord_le8783759336481291722_set_a @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_544_greaterThanLessThan__iff,axiom,
! [I: set_set_a,L: set_set_a,U: set_set_a] :
( ( member_set_set_a @ I @ ( set_or5544419673604628330_set_a @ L @ U ) )
= ( ( ord_less_set_set_a @ L @ I )
& ( ord_less_set_set_a @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_545_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_546_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_547_atLeastLessThan__iff,axiom,
! [I: set_set_set_a,L: set_set_set_a,U: set_set_set_a] :
( ( member_set_set_set_a @ I @ ( set_or4667047852671975087_set_a @ L @ U ) )
= ( ( ord_le5722252365846178494_set_a @ L @ I )
& ( ord_le8783759336481291722_set_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_548_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_549_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_550_atLeastLessThan__iff,axiom,
! [I: set_set_a,L: set_set_a,U: set_set_a] :
( ( member_set_set_a @ I @ ( set_or7717641506629902671_set_a @ L @ U ) )
= ( ( ord_le3724670747650509150_set_a @ L @ I )
& ( ord_less_set_set_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_551_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_552_atMost__subset__iff,axiom,
! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le8049040685576063006_set_a @ ( set_or3788817117504262629_set_a @ X ) @ ( set_or3788817117504262629_set_a @ Y2 ) )
= ( ord_le5722252365846178494_set_a @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_553_atMost__subset__iff,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4016371710855203973_set_a @ X ) @ ( set_or4016371710855203973_set_a @ Y2 ) )
= ( ord_le3724670747650509150_set_a @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_554_atMost__subset__iff,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ X ) @ ( set_ord_atMost_a @ Y2 ) )
= ( ord_less_eq_a @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_555_atMost__subset__iff,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X ) @ ( set_ord_atMost_set_a @ Y2 ) )
= ( ord_less_eq_set_a @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_556_greaterThanAtMost__iff,axiom,
! [I: set_set_set_a,L: set_set_set_a,U: set_set_set_a] :
( ( member_set_set_set_a @ I @ ( set_or5616217913275155566_set_a @ L @ U ) )
= ( ( ord_le8783759336481291722_set_a @ L @ I )
& ( ord_le5722252365846178494_set_a @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_557_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_558_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_559_greaterThanAtMost__iff,axiom,
! [I: set_set_a,L: set_set_a,U: set_set_a] :
( ( member_set_set_a @ I @ ( set_or6073455820589966094_set_a @ L @ U ) )
= ( ( ord_less_set_set_a @ L @ I )
& ( ord_le3724670747650509150_set_a @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_560_psubsetD,axiom,
! [A: set_set_set_set_a,B: set_set_set_set_a,C: set_set_set_a] :
( ( ord_le6641866341576993578_set_a @ A @ B )
=> ( ( member_set_set_set_a @ C @ A )
=> ( member_set_set_set_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_561_psubsetD,axiom,
! [A: set_set_set_a,B: set_set_set_a,C: set_set_a] :
( ( ord_le8783759336481291722_set_a @ A @ B )
=> ( ( member_set_set_a @ C @ A )
=> ( member_set_set_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_562_psubsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_563_psubsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_564_psubset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( ord_less_set_set_a @ B @ C2 )
=> ( ord_less_set_set_a @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_565_psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_566_psubset__trans,axiom,
! [A: set_set_set_a,B: set_set_set_a,C2: set_set_set_a] :
( ( ord_le8783759336481291722_set_a @ A @ B )
=> ( ( ord_le8783759336481291722_set_a @ B @ C2 )
=> ( ord_le8783759336481291722_set_a @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_567_atLeastLessThan__inj_I2_J,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ C @ D )
=> ( B2 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_568_atLeastLessThan__inj_I1_J,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ C @ D )
=> ( A2 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_569_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_570_atLeastLessThan__eq__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ C @ D )
=> ( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D ) )
= ( ( A2 = C )
& ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_571_Ioc__inj,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( set_or4472690218693186638Most_a @ A2 @ B2 )
= ( set_or4472690218693186638Most_a @ C @ D ) )
= ( ( ( ord_less_eq_a @ B2 @ A2 )
& ( ord_less_eq_a @ D @ C ) )
| ( ( A2 = C )
& ( B2 = D ) ) ) ) ).
% Ioc_inj
thf(fact_572_closed__atMost,axiom,
! [A2: a] : ( topolo784654279908865136osed_a @ ( set_ord_atMost_a @ A2 ) ) ).
% closed_atMost
thf(fact_573_atLeastLessThan__subset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ A2 @ B2 ) @ ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_eq_a @ B2 @ A2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_574_Ioc__subset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ A2 @ B2 ) @ ( set_or4472690218693186638Most_a @ C @ D ) )
= ( ( ord_less_eq_a @ B2 @ A2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_atLeastAtMost__def,axiom,
( set_or4761464488706262899_set_a
= ( ^ [L3: set_set_a,U2: set_set_a] : ( inf_in1205276777018777868_set_a @ ( set_or3904034815786525833_set_a @ L3 ) @ ( set_or4016371710855203973_set_a @ U2 ) ) ) ) ).
% atLeastAtMost_def
thf(fact_583_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_584_atLeastAtMost__def,axiom,
( set_or6288561110385358355_set_a
= ( ^ [L3: set_a,U2: set_a] : ( inf_inf_set_set_a @ ( set_or8362275514725411625_set_a @ L3 ) @ ( set_ord_atMost_set_a @ U2 ) ) ) ) ).
% atLeastAtMost_def
thf(fact_585_IntI,axiom,
! [C: set_set_set_a,A: set_set_set_set_a,B: set_set_set_set_a] :
( ( member_set_set_set_a @ C @ A )
=> ( ( member_set_set_set_a @ C @ B )
=> ( member_set_set_set_a @ C @ ( inf_in4918978645039662700_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_586_IntI,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ A )
=> ( ( member_set_set_a @ C @ B )
=> ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_587_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_588_IntI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_589_Int__iff,axiom,
! [C: set_set_set_a,A: set_set_set_set_a,B: set_set_set_set_a] :
( ( member_set_set_set_a @ C @ ( inf_in4918978645039662700_set_a @ A @ B ) )
= ( ( member_set_set_set_a @ C @ A )
& ( member_set_set_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_590_Int__iff,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( inf_in1205276777018777868_set_a @ A @ B ) )
= ( ( member_set_set_a @ C @ A )
& ( member_set_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_591_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_592_Int__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_593_UnCI,axiom,
! [C: set_set_set_a,B: set_set_set_set_a,A: set_set_set_set_a] :
( ( ~ ( member_set_set_set_a @ C @ B )
=> ( member_set_set_set_a @ C @ A ) )
=> ( member_set_set_set_a @ C @ ( sup_su6872963709084814930_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_594_UnCI,axiom,
! [C: set_set_a,B: set_set_set_a,A: set_set_set_a] :
( ( ~ ( member_set_set_a @ C @ B )
=> ( member_set_set_a @ C @ A ) )
=> ( member_set_set_a @ C @ ( sup_su2076012971530813682_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_595_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_596_UnCI,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ A ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_597_Un__iff,axiom,
! [C: set_set_set_a,A: set_set_set_set_a,B: set_set_set_set_a] :
( ( member_set_set_set_a @ C @ ( sup_su6872963709084814930_set_a @ A @ B ) )
= ( ( member_set_set_set_a @ C @ A )
| ( member_set_set_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_598_Un__iff,axiom,
! [C: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ C @ ( sup_su2076012971530813682_set_a @ A @ B ) )
= ( ( member_set_set_a @ C @ A )
| ( member_set_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_599_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_600_Un__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
| ( member_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_601_Int__subset__iff,axiom,
! [C2: set_set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C2 @ ( inf_in1205276777018777868_set_a @ A @ B ) )
= ( ( ord_le5722252365846178494_set_a @ C2 @ A )
& ( ord_le5722252365846178494_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_602_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_603_Int__subset__iff,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A )
& ( ord_le3724670747650509150_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_604_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_605_Un__subset__iff,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 )
= ( ( ord_le3724670747650509150_set_a @ A @ C2 )
& ( ord_le3724670747650509150_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_606_closed__Int,axiom,
! [S: set_a,T3: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( topolo784654279908865136osed_a @ ( inf_inf_set_a @ S @ T3 ) ) ) ) ).
% closed_Int
thf(fact_607_closed__Un,axiom,
! [S: set_a,T3: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( topolo784654279908865136osed_a @ ( sup_sup_set_a @ S @ T3 ) ) ) ) ).
% closed_Un
thf(fact_608_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_609_UnE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
=> ( ~ ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% UnE
thf(fact_610_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_611_IntE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_612_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_613_UnI1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_614_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_615_UnI2,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_616_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_617_IntD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% IntD1
thf(fact_618_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_619_IntD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_620_Un__Int__assoc__eq,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_621_Un__Int__assoc__eq,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) ) )
= ( ord_le3724670747650509150_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_622_Un__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_623_Un__mono,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_624_Un__least,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_625_Un__least,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_626_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_627_Un__upper1,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_628_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_629_Un__upper2,axiom,
! [B: set_set_a,A: set_set_a] : ( ord_le3724670747650509150_set_a @ B @ ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_630_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_631_Un__absorb1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( sup_sup_set_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_632_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_633_Un__absorb2,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( sup_sup_set_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_634_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B )
=> ( C2
!= ( sup_sup_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_635_subset__UnE,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) )
=> ~ ! [A7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ A )
=> ! [B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B7 @ B )
=> ( C2
!= ( sup_sup_set_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_636_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_637_subset__Un__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( sup_sup_set_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_638_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_639_Int__mono,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_640_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_641_Int__lower1,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_642_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_643_Int__lower2,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_644_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_645_Int__absorb1,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( inf_inf_set_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_646_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_647_Int__absorb2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( inf_inf_set_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_648_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_649_Int__greatest,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_650_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_651_Int__Collect__mono,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_652_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_653_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_654_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_655_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_656_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_657_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_658_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_659_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_660_sup_Obounded__iff,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 )
= ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
& ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_661_le__sup__iff,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_662_le__sup__iff,axiom,
! [X: set_set_a,Y2: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X @ Y2 ) @ Z2 )
= ( ( ord_le3724670747650509150_set_a @ X @ Z2 )
& ( ord_le3724670747650509150_set_a @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_663_inf_Obounded__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_664_inf_Obounded__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_665_le__inf__iff,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y2 )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_666_le__inf__iff,axiom,
! [X: set_set_a,Y2: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y2 @ Z2 ) )
= ( ( ord_le3724670747650509150_set_a @ X @ Y2 )
& ( ord_le3724670747650509150_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_667_distrib__sup__le,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y2 ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_668_distrib__sup__le,axiom,
! [X: set_set_a,Y2: set_set_a,Z2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X @ ( inf_inf_set_set_a @ Y2 @ Z2 ) ) @ ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X @ Y2 ) @ ( sup_sup_set_set_a @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_669_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_670_inf__sup__ord_I2_J,axiom,
! [X: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_671_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_672_inf__sup__ord_I1_J,axiom,
! [X: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_673_inf__le1,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_674_inf__le1,axiom,
! [X: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_675_inf__le2,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_676_inf__le2,axiom,
! [X: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_677_le__infE,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A2 )
=> ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_678_le__infE,axiom,
! [X: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X @ A2 )
=> ~ ( ord_le3724670747650509150_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_679_le__infI,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_680_le__infI,axiom,
! [X: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X @ B2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_681_inf__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_682_inf__mono,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_683_le__infI1,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_684_le__infI1,axiom,
! [A2: set_set_a,X: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_685_le__infI2,axiom,
! [B2: set_a,X: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_686_le__infI2,axiom,
! [B2: set_set_a,X: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_687_inf_OorderE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_688_inf_OorderE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_689_inf_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_690_inf_OorderI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2
= ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_691_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y2: set_a] :
( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_a,Y4: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( ord_less_eq_set_a @ X2 @ Z4 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_692_inf__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X: set_set_a,Y2: set_set_a] :
( ! [X2: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_set_a,Y4: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y4 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Z4 )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_set_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_693_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_694_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_695_inf_Oabsorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_696_inf_Oabsorb1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_697_inf_Oabsorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_698_inf_Oabsorb2,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_699_inf__absorb1,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( inf_inf_set_a @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_700_inf__absorb1,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( inf_inf_set_set_a @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_701_inf__absorb2,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( inf_inf_set_a @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_702_inf__absorb2,axiom,
! [Y2: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X )
=> ( ( inf_inf_set_set_a @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_703_inf_OboundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_704_inf_OboundedE,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_705_inf_OboundedI,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_706_inf_OboundedI,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_707_inf__greatest,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_708_inf__greatest,axiom,
! [X: set_set_a,Y2: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_709_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_710_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( A4
= ( inf_inf_set_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_711_inf_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_712_inf_Ocobounded1,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_713_inf_Ocobounded2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_714_inf_Ocobounded2,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_715_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_716_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_717_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_718_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_719_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_720_inf_OcoboundedI1,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_721_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_722_inf_OcoboundedI2,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_723_inf__sup__ord_I4_J,axiom,
! [Y2: set_a,X: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_724_inf__sup__ord_I4_J,axiom,
! [Y2: set_set_a,X: set_set_a] : ( ord_le3724670747650509150_set_a @ Y2 @ ( sup_sup_set_set_a @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_725_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_726_inf__sup__ord_I3_J,axiom,
! [X: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_727_le__supE,axiom,
! [A2: set_a,B2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X )
=> ~ ( ord_less_eq_set_a @ B2 @ X ) ) ) ).
% le_supE
thf(fact_728_le__supE,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ X )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ X ) ) ) ).
% le_supE
thf(fact_729_le__supI,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_730_le__supI,axiom,
! [A2: set_set_a,X: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ X )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ X )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_731_sup__ge1,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_732_sup__ge1,axiom,
! [X: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_733_sup__ge2,axiom,
! [Y2: set_a,X: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_734_sup__ge2,axiom,
! [Y2: set_set_a,X: set_set_a] : ( ord_le3724670747650509150_set_a @ Y2 @ ( sup_sup_set_set_a @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_735_le__supI1,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_736_le__supI1,axiom,
! [X: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_737_le__supI2,axiom,
! [X: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_738_le__supI2,axiom,
! [X: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ B2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_739_sup_Omono,axiom,
! [C: set_a,A2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_740_sup_Omono,axiom,
! [C: set_set_a,A2: set_set_a,D: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ D @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ C @ D ) @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_741_sup__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_742_sup__mono,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( sup_sup_set_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_743_sup__least,axiom,
! [Y2: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_744_sup__least,axiom,
! [Y2: set_set_a,X: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X )
=> ( ( ord_le3724670747650509150_set_a @ Z2 @ X )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_745_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_746_le__iff__sup,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_747_sup_OorderE,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_748_sup_OorderE,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_749_sup_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_750_sup_OorderI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2
= ( sup_sup_set_set_a @ A2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_751_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y2: set_a] :
( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_a,Y4: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( ( ord_less_eq_set_a @ Z4 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y4 @ Z4 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_752_sup__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X: set_set_a,Y2: set_set_a] :
( ! [X2: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_set_a,Y4: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y4 @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ Z4 @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ Y4 @ Z4 ) @ X2 ) ) )
=> ( ( sup_sup_set_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_753_sup_Oabsorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_754_sup_Oabsorb1,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_755_sup_Oabsorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_756_sup_Oabsorb2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_757_sup__absorb1,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( sup_sup_set_a @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_758_sup__absorb1,axiom,
! [Y2: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X )
=> ( ( sup_sup_set_set_a @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_759_sup__absorb2,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( sup_sup_set_a @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_760_sup__absorb2,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( sup_sup_set_set_a @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_761_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_762_sup_OboundedE,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ~ ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_763_sup_OboundedI,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_764_sup_OboundedI,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_765_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_766_sup_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( A4
= ( sup_sup_set_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_767_sup_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_768_sup_Ocobounded1,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_769_sup_Ocobounded2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_770_sup_Ocobounded2,axiom,
! [B2: set_set_a,A2: set_set_a] : ( ord_le3724670747650509150_set_a @ B2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_771_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_772_sup_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_773_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_774_sup_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_775_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_776_sup_OcoboundedI1,axiom,
! [C: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_777_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_778_sup_OcoboundedI2,axiom,
! [C: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_779_distrib__inf__le,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ ( inf_inf_set_a @ X @ Z2 ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_780_distrib__inf__le,axiom,
! [X: set_set_a,Y2: set_set_a,Z2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X @ Y2 ) @ ( inf_inf_set_set_a @ X @ Z2 ) ) @ ( inf_inf_set_set_a @ X @ ( sup_sup_set_set_a @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_781_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_782_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_783_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_784_lessThan__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_ord_lessThan_a @ K ) )
= ( ord_less_a @ I @ K ) ) ).
% lessThan_iff
thf(fact_785_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_786_greaterThan__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_or8632414552788122084Than_a @ K ) )
= ( ord_less_a @ K @ I ) ) ).
% greaterThan_iff
thf(fact_787_lessThan__subset__iff,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_set_a @ ( set_ord_lessThan_a @ X ) @ ( set_ord_lessThan_a @ Y2 ) )
= ( ord_less_eq_a @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_788_greaterThan__subset__iff,axiom,
! [X: a,Y2: a] :
( ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ X ) @ ( set_or8632414552788122084Than_a @ Y2 ) )
= ( ord_less_eq_a @ Y2 @ X ) ) ).
% greaterThan_subset_iff
thf(fact_789_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_790_Ioi__le__Ico,axiom,
! [A2: a] : ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ A2 ) @ ( set_ord_atLeast_a @ A2 ) ) ).
% Ioi_le_Ico
thf(fact_791_Ioi__le__Ico,axiom,
! [A2: set_a] : ( ord_le3724670747650509150_set_a @ ( set_or460448635090783044_set_a @ A2 ) @ ( set_or8362275514725411625_set_a @ A2 ) ) ).
% Ioi_le_Ico
thf(fact_792_Iic__subset__Iio__iff,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ A2 ) @ ( set_ord_lessThan_a @ B2 ) )
= ( ord_less_a @ A2 @ B2 ) ) ).
% Iic_subset_Iio_iff
thf(fact_793_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_794_Ici__subset__Ioi__iff,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ A2 ) @ ( set_or8632414552788122084Than_a @ B2 ) )
= ( ord_less_a @ B2 @ A2 ) ) ).
% Ici_subset_Ioi_iff
thf(fact_795_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_796_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_797_atLeastLessThan__def,axiom,
( set_or2348907005316661231_set_a
= ( ^ [L3: set_a,U2: set_a] : ( inf_inf_set_set_a @ ( set_or8362275514725411625_set_a @ L3 ) @ ( set_or5421148953861284865_set_a @ U2 ) ) ) ) ).
% atLeastLessThan_def
thf(fact_798_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_799_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_800_less__separate,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_set_a @ X @ Y2 )
=> ? [A3: set_a,B3: set_a] :
( ( member_set_a @ X @ ( set_or5421148953861284865_set_a @ A3 ) )
& ( member_set_a @ Y2 @ ( set_or460448635090783044_set_a @ B3 ) )
& ( ( inf_inf_set_set_a @ ( set_or5421148953861284865_set_a @ A3 ) @ ( set_or460448635090783044_set_a @ B3 ) )
= bot_bot_set_set_a ) ) ) ).
% less_separate
thf(fact_801_less__separate,axiom,
! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ? [A3: a,B3: a] :
( ( member_a @ X @ ( set_ord_lessThan_a @ A3 ) )
& ( member_a @ Y2 @ ( set_or8632414552788122084Than_a @ B3 ) )
& ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ A3 ) @ ( set_or8632414552788122084Than_a @ B3 ) )
= bot_bot_set_a ) ) ) ).
% less_separate
thf(fact_802_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_803_ivl__disj__int__one_I6_J,axiom,
! [L: set_a,U: set_a] :
( ( inf_inf_set_set_a @ ( set_or6017932776736107018_set_a @ L @ U ) @ ( set_or8362275514725411625_set_a @ U ) )
= bot_bot_set_set_a ) ).
% ivl_disj_int_one(6)
thf(fact_804_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_805_UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_806_atLeast__empty__triv,axiom,
( ( set_or8362275514725411625_set_a @ bot_bot_set_a )
= top_top_set_set_a ) ).
% atLeast_empty_triv
thf(fact_807_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_808_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_809_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_810_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_811_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_812_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_813_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_814_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_815_closed__UNIV,axiom,
topolo784654279908865136osed_a @ top_top_set_a ).
% closed_UNIV
thf(fact_816_closed__empty,axiom,
topolo784654279908865136osed_a @ bot_bot_set_a ).
% closed_empty
thf(fact_817_atLeastatMost__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( set_or6288561110385358355_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_818_atLeastatMost__empty__iff,axiom,
! [A2: a,B2: a] :
( ( ( set_or672772299803893939Most_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ~ ( ord_less_eq_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_819_atLeastatMost__empty__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( set_or4761464488706262899_set_a @ A2 @ B2 )
= bot_bo3380559777022489994_set_a )
= ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_820_atLeastatMost__empty__iff2,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_set_a
= ( set_or6288561110385358355_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_821_atLeastatMost__empty__iff2,axiom,
! [A2: a,B2: a] :
( ( bot_bot_set_a
= ( set_or672772299803893939Most_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_822_atLeastatMost__empty__iff2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( bot_bo3380559777022489994_set_a
= ( set_or4761464488706262899_set_a @ A2 @ B2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_823_atLeastatMost__empty_H,axiom,
! [A2: set_a,B2: set_a] :
( ~ ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( set_or6288561110385358355_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% atLeastatMost_empty'
thf(fact_824_atLeastatMost__empty_H,axiom,
! [A2: a,B2: a] :
( ~ ( ord_less_eq_a @ A2 @ B2 )
=> ( ( set_or672772299803893939Most_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% atLeastatMost_empty'
thf(fact_825_atLeastatMost__empty_H,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( set_or4761464488706262899_set_a @ A2 @ B2 )
= bot_bo3380559777022489994_set_a ) ) ).
% atLeastatMost_empty'
thf(fact_826_atLeastatMost__empty,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ( set_or672772299803893939Most_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% atLeastatMost_empty
thf(fact_827_atLeastLessThan__empty,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( set_or2348907005316661231_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_828_atLeastLessThan__empty,axiom,
! [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_829_atLeastLessThan__empty,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( set_or7717641506629902671_set_a @ A2 @ B2 )
= bot_bo3380559777022489994_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_830_atLeastLessThan__empty__iff2,axiom,
! [A2: a,B2: a] :
( ( bot_bot_set_a
= ( set_or5139330845457685135Than_a @ A2 @ B2 ) )
= ( ~ ( ord_less_a @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_831_atLeastLessThan__empty__iff,axiom,
! [A2: a,B2: a] :
( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ~ ( ord_less_a @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_832_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_833_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_834_greaterThanLessThan__empty,axiom,
! [L: set_set_a,K: set_set_a] :
( ( ord_le3724670747650509150_set_a @ L @ K )
=> ( ( set_or5544419673604628330_set_a @ K @ L )
= bot_bo3380559777022489994_set_a ) ) ).
% greaterThanLessThan_empty
thf(fact_835_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_836_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_837_greaterThanAtMost__empty,axiom,
! [L: set_set_a,K: set_set_a] :
( ( ord_le3724670747650509150_set_a @ L @ K )
=> ( ( set_or6073455820589966094_set_a @ K @ L )
= bot_bo3380559777022489994_set_a ) ) ).
% greaterThanAtMost_empty
thf(fact_838_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_839_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_840_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_841_bot_Oextremum,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% bot.extremum
thf(fact_842_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_843_bot_Oextremum__unique,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_844_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_845_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
=> ( A2 = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_846_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_847_top__greatest,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ top_top_set_set_a ) ).
% top_greatest
thf(fact_848_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_849_top_Oextremum__unique,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A2 )
= ( A2 = top_top_set_set_a ) ) ).
% top.extremum_unique
thf(fact_850_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_851_top_Oextremum__uniqueI,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A2 )
=> ( A2 = top_top_set_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_852_atLeast__eq__UNIV__iff,axiom,
! [X: set_a] :
( ( ( set_or8362275514725411625_set_a @ X )
= top_top_set_set_a )
= ( X = bot_bot_set_a ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_853_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_854_UNIV__witness,axiom,
? [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_855_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_856_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_857_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_858_UNIV__eq__I,axiom,
! [A: set_set_a] :
( ! [X2: set_a] : ( member_set_a @ X2 @ A )
=> ( top_top_set_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_859_equals0I,axiom,
! [A: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_860_equals0I,axiom,
! [A: set_set_a] :
( ! [Y4: set_a] :
~ ( member_set_a @ Y4 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_861_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_862_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_863_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_864_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_865_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_866_subset__UNIV,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ top_top_set_set_a ) ).
% subset_UNIV
thf(fact_867_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_868_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_869_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_870_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ~ ( member_set_a @ X2 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_871_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_872_not__empty__eq__Ici__eq__empty,axiom,
! [L: set_a] :
( bot_bot_set_set_a
!= ( set_or8362275514725411625_set_a @ L ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_873_Ioc__disjoint,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( inf_inf_set_a @ ( set_or4472690218693186638Most_a @ A2 @ B2 ) @ ( set_or4472690218693186638Most_a @ C @ D ) )
= bot_bot_set_a )
= ( ( ord_less_eq_a @ B2 @ A2 )
| ( ord_less_eq_a @ D @ C )
| ( ord_less_eq_a @ B2 @ C )
| ( ord_less_eq_a @ D @ A2 ) ) ) ).
% Ioc_disjoint
thf(fact_874_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_875_ivl__disj__int__one_I8_J,axiom,
! [L: set_a,U: set_a] :
( ( inf_inf_set_set_a @ ( set_or2348907005316661231_set_a @ L @ U ) @ ( set_or8362275514725411625_set_a @ U ) )
= bot_bot_set_set_a ) ).
% ivl_disj_int_one(8)
thf(fact_876_subset__emptyI,axiom,
! [A: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_877_subset__emptyI,axiom,
! [A: set_set_a] :
( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_878_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_879_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila844772492610406764_set_a @ inf_inf_set_set_a @ top_top_set_set_a @ ord_le3724670747650509150_set_a @ ord_less_set_set_a ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_880_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_881_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_882_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_883_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_884_insertCI,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_885_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_886_insert__iff,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_887_insert__subset,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_888_insert__subset,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A ) @ B )
= ( ( member_set_a @ X @ B )
& ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_889_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_890_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_891_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_892_Int__insert__left__if0,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_893_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_894_Int__insert__left__if1,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_895_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_896_Int__insert__right__if0,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_897_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_898_Int__insert__right__if1,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_899_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_900_singleton__insert__inj__eq,axiom,
! [B2: set_a,A2: set_a,A: set_set_a] :
( ( ( insert_set_a @ B2 @ bot_bot_set_set_a )
= ( insert_set_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_901_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B2: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_902_singleton__insert__inj__eq_H,axiom,
! [A2: set_a,A: set_set_a,B2: set_a] :
( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( ( A2 = B2 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_903_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_904_insert__disjoint_I1_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_905_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_906_insert__disjoint_I2_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B ) )
= ( ~ ( member_set_a @ A2 @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_907_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_908_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A2: set_a,A: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A2 @ A ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ B @ A )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_909_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_910_disjoint__insert_I2_J,axiom,
! [A: set_set_a,B2: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ ( insert_set_a @ B2 @ B ) ) )
= ( ~ ( member_set_a @ B2 @ A )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_911_closed__singleton,axiom,
! [A2: a] : ( topolo784654279908865136osed_a @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% closed_singleton
thf(fact_912_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_913_singleton__iff,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_914_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_915_singletonD,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_916_closed__insert,axiom,
! [S: set_a,A2: a] :
( ( topolo784654279908865136osed_a @ S )
=> ( topolo784654279908865136osed_a @ ( insert_a @ A2 @ S ) ) ) ).
% closed_insert
thf(fact_917_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_918_insertE,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_919_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_920_insertI1,axiom,
! [A2: set_a,B: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B ) ) ).
% insertI1
thf(fact_921_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_922_insertI2,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_923_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B8: set_a] :
( ( A
= ( insert_a @ X @ B8 ) )
=> ( member_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_924_Set_Oset__insert,axiom,
! [X: set_a,A: set_set_a] :
( ( member_set_a @ X @ A )
=> ~ ! [B8: set_set_a] :
( ( A
= ( insert_set_a @ X @ B8 ) )
=> ( member_set_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_925_insert__ident,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A )
= ( insert_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_926_insert__ident,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( ( insert_set_a @ X @ A )
= ( insert_set_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_927_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_928_insert__absorb,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_929_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_930_insert__eq__iff,axiom,
! [A2: set_a,A: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_a] :
( ( A
= ( insert_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ B2 @ C3 )
& ( B
= ( insert_set_a @ A2 @ C3 ) )
& ~ ( member_set_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_931_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B8: set_a] :
( ( A
= ( insert_a @ A2 @ B8 ) )
& ~ ( member_a @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_932_mk__disjoint__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ? [B8: set_set_a] :
( ( A
= ( insert_set_a @ A2 @ B8 ) )
& ~ ( member_set_a @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_933_insert__subsetI,axiom,
! [X: a,A: set_a,X5: set_a] :
( ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X5 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_934_insert__subsetI,axiom,
! [X: set_a,A: set_set_a,X5: set_set_a] :
( ( member_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_935_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_936_Int__insert__right,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) )
& ( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_937_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_938_Int__insert__left,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_939_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_940_subset__insertI2,axiom,
! [A: set_set_a,B: set_set_a,B2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_941_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_942_subset__insertI,axiom,
! [B: set_set_a,A2: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_943_subset__insert,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_944_subset__insert,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X @ B ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_945_insert__mono,axiom,
! [C2: set_a,D2: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_946_insert__mono,axiom,
! [C2: set_set_a,D2: set_set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A2 @ C2 ) @ ( insert_set_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_947_subset__singletonD,axiom,
! [A: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_948_subset__singletonD,axiom,
! [A: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
=> ( ( A = bot_bot_set_set_a )
| ( A
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_949_subset__singleton__iff,axiom,
! [X5: set_a,A2: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_950_subset__singleton__iff,axiom,
! [X5: set_set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( ( X5 = bot_bot_set_set_a )
| ( X5
= ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_951_Iio__Int__singleton,axiom,
! [X: a,K: a] :
( ( ( ord_less_a @ X @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( insert_a @ X @ bot_bot_set_a ) ) )
& ( ~ ( ord_less_a @ X @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X @ bot_bot_set_a ) )
= bot_bot_set_a ) ) ) ).
% Iio_Int_singleton
thf(fact_952_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_953_atMost__Int__atLeast,axiom,
! [N: set_a] :
( ( inf_inf_set_set_a @ ( set_ord_atMost_set_a @ N ) @ ( set_or8362275514725411625_set_a @ N ) )
= ( insert_set_a @ N @ bot_bot_set_set_a ) ) ).
% atMost_Int_atLeast
thf(fact_954_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_955_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_956_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_957_top_Oordering__top__axioms,axiom,
orderi5875812994216768367_set_a @ ord_le3724670747650509150_set_a @ ord_less_set_set_a @ top_top_set_set_a ).
% top.ordering_top_axioms
thf(fact_958_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y4 @ A )
=> ( X2 = Y4 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_959_is__singletonI_H,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y4: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( member_set_a @ Y4 @ A )
=> ( X2 = Y4 ) ) )
=> ( is_singleton_set_a @ A ) ) ) ).
% is_singletonI'
thf(fact_960_connected__closed,axiom,
( topolo2370605967727889109cted_a
= ( ^ [S2: set_a] :
~ ? [A5: set_a,B5: set_a] :
( ( topolo784654279908865136osed_a @ A5 )
& ( topolo784654279908865136osed_a @ B5 )
& ( ord_less_eq_set_a @ S2 @ ( sup_sup_set_a @ A5 @ B5 ) )
& ( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ S2 )
= bot_bot_set_a )
& ( ( inf_inf_set_a @ A5 @ S2 )
!= bot_bot_set_a )
& ( ( inf_inf_set_a @ B5 @ S2 )
!= bot_bot_set_a ) ) ) ) ).
% connected_closed
thf(fact_961_connected__closedD,axiom,
! [S3: set_a,A: set_a,B: set_a] :
( ( topolo2370605967727889109cted_a @ S3 )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ S3 )
= bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ S3 @ ( sup_sup_set_a @ A @ B ) )
=> ( ( topolo784654279908865136osed_a @ A )
=> ( ( topolo784654279908865136osed_a @ B )
=> ( ( ( inf_inf_set_a @ A @ S3 )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ S3 )
= bot_bot_set_a ) ) ) ) ) ) ) ).
% connected_closedD
thf(fact_962_closed__Inter,axiom,
! [K2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ K2 )
=> ( topolo784654279908865136osed_a @ X2 ) )
=> ( topolo784654279908865136osed_a @ ( comple6135023378680113637_set_a @ K2 ) ) ) ).
% closed_Inter
thf(fact_963_Inf__atLeast,axiom,
! [X: set_a] :
( ( comple6135023378680113637_set_a @ ( set_or8362275514725411625_set_a @ X ) )
= X ) ).
% Inf_atLeast
thf(fact_964_cInf__atLeast,axiom,
! [X: set_a] :
( ( comple6135023378680113637_set_a @ ( set_or8362275514725411625_set_a @ X ) )
= X ) ).
% cInf_atLeast
thf(fact_965_cInf__atLeastAtMost,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ Y2 @ X ) )
= Y2 ) ) ).
% cInf_atLeastAtMost
thf(fact_966_cInf__atLeastAtMost,axiom,
! [Y2: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X )
=> ( ( comple9105089376463352645_set_a @ ( set_or4761464488706262899_set_a @ Y2 @ X ) )
= Y2 ) ) ).
% cInf_atLeastAtMost
thf(fact_967_Inf__atLeastAtMost,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ X @ Y2 ) )
= X ) ) ).
% Inf_atLeastAtMost
thf(fact_968_Inf__atLeastAtMost,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( comple9105089376463352645_set_a @ ( set_or4761464488706262899_set_a @ X @ Y2 ) )
= X ) ) ).
% Inf_atLeastAtMost
thf(fact_969_cInf__eq__minimum,axiom,
! [Z2: set_a,X5: set_set_a] :
( ( member_set_a @ Z2 @ X5 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ord_less_eq_set_a @ Z2 @ X2 ) )
=> ( ( comple6135023378680113637_set_a @ X5 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_970_cInf__eq__minimum,axiom,
! [Z2: set_set_a,X5: set_set_set_a] :
( ( member_set_set_a @ Z2 @ X5 )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ X5 )
=> ( ord_le3724670747650509150_set_a @ Z2 @ X2 ) )
=> ( ( comple9105089376463352645_set_a @ X5 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_971_connectedD__interval,axiom,
! [U3: set_a,X: a,Y2: a,Z2: a] :
( ( topolo2370605967727889109cted_a @ U3 )
=> ( ( member_a @ X @ U3 )
=> ( ( member_a @ Y2 @ U3 )
=> ( ( ord_less_eq_a @ X @ Z2 )
=> ( ( ord_less_eq_a @ Z2 @ Y2 )
=> ( member_a @ Z2 @ U3 ) ) ) ) ) ) ).
% connectedD_interval
thf(fact_972_cInf__eq__non__empty,axiom,
! [X5: set_set_a,A2: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ord_less_eq_set_a @ A2 @ X2 ) )
=> ( ! [Y4: set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ X5 )
=> ( ord_less_eq_set_a @ Y4 @ X4 ) )
=> ( ord_less_eq_set_a @ Y4 @ A2 ) )
=> ( ( comple6135023378680113637_set_a @ X5 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_973_cInf__eq__non__empty,axiom,
! [X5: set_set_set_a,A2: set_set_a] :
( ( X5 != bot_bo3380559777022489994_set_a )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ X5 )
=> ( ord_le3724670747650509150_set_a @ A2 @ X2 ) )
=> ( ! [Y4: set_set_a] :
( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ X5 )
=> ( ord_le3724670747650509150_set_a @ Y4 @ X4 ) )
=> ( ord_le3724670747650509150_set_a @ Y4 @ A2 ) )
=> ( ( comple9105089376463352645_set_a @ X5 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_974_cInf__greatest,axiom,
! [X5: set_set_a,Z2: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ord_less_eq_set_a @ Z2 @ X2 ) )
=> ( ord_less_eq_set_a @ Z2 @ ( comple6135023378680113637_set_a @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_975_cInf__greatest,axiom,
! [X5: set_set_set_a,Z2: set_set_a] :
( ( X5 != bot_bo3380559777022489994_set_a )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ X5 )
=> ( ord_le3724670747650509150_set_a @ Z2 @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ Z2 @ ( comple9105089376463352645_set_a @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_976_connected__contains__Icc,axiom,
! [A: set_a,A2: a,B2: a] :
( ( topolo2370605967727889109cted_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ A )
=> ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ A ) ) ) ) ).
% connected_contains_Icc
thf(fact_977_connected__contains__Ioo,axiom,
! [A: set_a,A2: a,B2: a] :
( ( topolo2370605967727889109cted_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ A )
=> ( ord_less_eq_set_a @ ( set_or5939364468397584554Than_a @ A2 @ B2 ) @ A ) ) ) ) ).
% connected_contains_Ioo
thf(fact_978_connected__closed__set,axiom,
! [S: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo2370605967727889109cted_a @ S )
= ( ~ ? [A5: set_a,B5: set_a] :
( ( topolo784654279908865136osed_a @ A5 )
& ( topolo784654279908865136osed_a @ B5 )
& ( A5 != bot_bot_set_a )
& ( B5 != bot_bot_set_a )
& ( ( sup_sup_set_a @ A5 @ B5 )
= S )
& ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a ) ) ) ) ) ).
% connected_closed_set
thf(fact_979_Inter__lower,axiom,
! [B: set_a,A: set_set_a] :
( ( member_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ B ) ) ).
% Inter_lower
thf(fact_980_Inter__lower,axiom,
! [B: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ B @ A )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A ) @ B ) ) ).
% Inter_lower
thf(fact_981_Inter__greatest,axiom,
! [A: set_set_a,C2: set_a] :
( ! [X6: set_a] :
( ( member_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ C2 @ X6 ) )
=> ( ord_less_eq_set_a @ C2 @ ( comple6135023378680113637_set_a @ A ) ) ) ).
% Inter_greatest
thf(fact_982_Inter__greatest,axiom,
! [A: set_set_set_a,C2: set_set_a] :
( ! [X6: set_set_a] :
( ( member_set_set_a @ X6 @ A )
=> ( ord_le3724670747650509150_set_a @ C2 @ X6 ) )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( comple9105089376463352645_set_a @ A ) ) ) ).
% Inter_greatest
thf(fact_983_Inter__anti__mono,axiom,
! [B: set_set_set_a,A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B @ A )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A ) @ ( comple9105089376463352645_set_a @ B ) ) ) ).
% Inter_anti_mono
thf(fact_984_Inter__anti__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inter_anti_mono
thf(fact_985_Inter__subset,axiom,
! [A: set_set_a,B: set_a] :
( ! [X6: set_a] :
( ( member_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ X6 @ B ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_986_Inter__subset,axiom,
! [A: set_set_set_a,B: set_set_a] :
( ! [X6: set_set_a] :
( ( member_set_set_a @ X6 @ A )
=> ( ord_le3724670747650509150_set_a @ X6 @ B ) )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_987_Inf__eqI,axiom,
! [A: set_set_a,X: set_a] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A )
=> ( ord_less_eq_set_a @ X @ I3 ) )
=> ( ! [Y4: set_a] :
( ! [I4: set_a] :
( ( member_set_a @ I4 @ A )
=> ( ord_less_eq_set_a @ Y4 @ I4 ) )
=> ( ord_less_eq_set_a @ Y4 @ X ) )
=> ( ( comple6135023378680113637_set_a @ A )
= X ) ) ) ).
% Inf_eqI
thf(fact_988_Inf__eqI,axiom,
! [A: set_set_set_a,X: set_set_a] :
( ! [I3: set_set_a] :
( ( member_set_set_a @ I3 @ A )
=> ( ord_le3724670747650509150_set_a @ X @ I3 ) )
=> ( ! [Y4: set_set_a] :
( ! [I4: set_set_a] :
( ( member_set_set_a @ I4 @ A )
=> ( ord_le3724670747650509150_set_a @ Y4 @ I4 ) )
=> ( ord_le3724670747650509150_set_a @ Y4 @ X ) )
=> ( ( comple9105089376463352645_set_a @ A )
= X ) ) ) ).
% Inf_eqI
thf(fact_989_Inf__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ! [B3: set_a] :
( ( member_set_a @ B3 @ B )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ X4 @ B3 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_mono
thf(fact_990_Inf__mono,axiom,
! [B: set_set_set_a,A: set_set_set_a] :
( ! [B3: set_set_a] :
( ( member_set_set_a @ B3 @ B )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
& ( ord_le3724670747650509150_set_a @ X4 @ B3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A ) @ ( comple9105089376463352645_set_a @ B ) ) ) ).
% Inf_mono
thf(fact_991_Inf__lower,axiom,
! [X: set_a,A: set_set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ X ) ) ).
% Inf_lower
thf(fact_992_Inf__lower,axiom,
! [X: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ X @ A )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A ) @ X ) ) ).
% Inf_lower
thf(fact_993_Inf__lower2,axiom,
! [U: set_a,A: set_set_a,V: set_a] :
( ( member_set_a @ U @ A )
=> ( ( ord_less_eq_set_a @ U @ V )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_994_Inf__lower2,axiom,
! [U: set_set_a,A: set_set_set_a,V: set_set_a] :
( ( member_set_set_a @ U @ A )
=> ( ( ord_le3724670747650509150_set_a @ U @ V )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_995_le__Inf__iff,axiom,
! [B2: set_a,A: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( comple6135023378680113637_set_a @ A ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_996_le__Inf__iff,axiom,
! [B2: set_set_a,A: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( comple9105089376463352645_set_a @ A ) )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
=> ( ord_le3724670747650509150_set_a @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_997_Inf__greatest,axiom,
! [A: set_set_a,Z2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ Z2 @ X2 ) )
=> ( ord_less_eq_set_a @ Z2 @ ( comple6135023378680113637_set_a @ A ) ) ) ).
% Inf_greatest
thf(fact_998_Inf__greatest,axiom,
! [A: set_set_set_a,Z2: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ Z2 @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ Z2 @ ( comple9105089376463352645_set_a @ A ) ) ) ).
% Inf_greatest
thf(fact_999_Inter__Un__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B ) ) ) ).
% Inter_Un_subset
thf(fact_1000_Inter__Un__subset,axiom,
! [A: set_set_set_a,B: set_set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( comple9105089376463352645_set_a @ A ) @ ( comple9105089376463352645_set_a @ B ) ) @ ( comple9105089376463352645_set_a @ ( inf_in1205276777018777868_set_a @ A @ B ) ) ) ).
% Inter_Un_subset
thf(fact_1001_Inf__superset__mono,axiom,
! [B: set_set_set_a,A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B @ A )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A ) @ ( comple9105089376463352645_set_a @ B ) ) ) ).
% Inf_superset_mono
thf(fact_1002_Inf__superset__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_superset_mono
thf(fact_1003_Inf__less__eq,axiom,
! [A: set_set_a,U: set_a] :
( ! [V2: set_a] :
( ( member_set_a @ V2 @ A )
=> ( ord_less_eq_set_a @ V2 @ U ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1004_Inf__less__eq,axiom,
! [A: set_set_set_a,U: set_set_a] :
( ! [V2: set_set_a] :
( ( member_set_set_a @ V2 @ A )
=> ( ord_le3724670747650509150_set_a @ V2 @ U ) )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1005_less__eq__Inf__inter,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B ) ) ) ).
% less_eq_Inf_inter
thf(fact_1006_less__eq__Inf__inter,axiom,
! [A: set_set_set_a,B: set_set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( comple9105089376463352645_set_a @ A ) @ ( comple9105089376463352645_set_a @ B ) ) @ ( comple9105089376463352645_set_a @ ( inf_in1205276777018777868_set_a @ A @ B ) ) ) ).
% less_eq_Inf_inter
thf(fact_1007_connected__as__closed__union,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( topolo2370605967727889109cted_a @ C2 )
=> ( ( C2
= ( sup_sup_set_a @ A @ B ) )
=> ( ( topolo784654279908865136osed_a @ A )
=> ( ( topolo784654279908865136osed_a @ B )
=> ( ( A != bot_bot_set_a )
=> ( ( B != bot_bot_set_a )
=> ( ( inf_inf_set_a @ A @ B )
!= bot_bot_set_a ) ) ) ) ) ) ) ).
% connected_as_closed_union
thf(fact_1008_chains__extend,axiom,
! [C: set_set_a,S: set_set_a,Z2: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ Z2 @ S )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ C )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z2 @ bot_bot_set_set_a ) @ C ) @ ( chains_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1009_chains__extend,axiom,
! [C: set_set_set_a,S: set_set_set_a,Z2: set_set_a] :
( ( member_set_set_set_a @ C @ ( chains_set_a @ S ) )
=> ( ( member_set_set_a @ Z2 @ S )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ C )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z2 ) )
=> ( member_set_set_set_a @ ( sup_su2076012971530813682_set_a @ ( insert_set_set_a @ Z2 @ bot_bo3380559777022489994_set_a ) @ C ) @ ( chains_set_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1010_connectedD,axiom,
! [A: set_a,U3: set_a,V3: set_a] :
( ( topolo2370605967727889109cted_a @ A )
=> ( ( topolo8477419352202985285open_a @ U3 )
=> ( ( topolo8477419352202985285open_a @ V3 )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ U3 @ V3 ) @ A )
= bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ U3 @ V3 ) )
=> ( ( ( inf_inf_set_a @ U3 @ A )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ V3 @ A )
= bot_bot_set_a ) ) ) ) ) ) ) ).
% connectedD
thf(fact_1011_mono__atLeast,axiom,
! [B: a] : ( extend2808419353335425523_set_a @ ( set_ord_atLeast_a @ B ) ) ).
% mono_atLeast
thf(fact_1012_mono__atLeast,axiom,
! [B: set_a] : ( extend347329919781519187_set_a @ ( set_or8362275514725411625_set_a @ B ) ) ).
% mono_atLeast
thf(fact_1013_open__subopen,axiom,
( topolo8477419352202985285open_a
= ( ^ [S4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ S4 )
=> ? [T4: set_a] :
( ( topolo8477419352202985285open_a @ T4 )
& ( member_a @ X3 @ T4 )
& ( ord_less_eq_set_a @ T4 @ S4 ) ) ) ) ) ).
% open_subopen
thf(fact_1014_topological__space__class_OopenI,axiom,
! [S: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ S )
=> ? [T5: set_a] :
( ( topolo8477419352202985285open_a @ T5 )
& ( member_a @ X2 @ T5 )
& ( ord_less_eq_set_a @ T5 @ S ) ) )
=> ( topolo8477419352202985285open_a @ S ) ) ).
% topological_space_class.openI
thf(fact_1015_Zorn__Lemma2,axiom,
! [A: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( chains_a @ A ) )
=> ? [Xa: set_a] :
( ( member_set_a @ Xa @ A )
& ! [Xb: set_a] :
( ( member_set_a @ Xb @ X2 )
=> ( ord_less_eq_set_a @ Xb @ Xa ) ) ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1016_Zorn__Lemma2,axiom,
! [A: set_set_set_a] :
( ! [X2: set_set_set_a] :
( ( member_set_set_set_a @ X2 @ ( chains_set_a @ A ) )
=> ? [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
& ! [Xb: set_set_a] :
( ( member_set_set_a @ Xb @ X2 )
=> ( ord_le3724670747650509150_set_a @ Xb @ Xa ) ) ) )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1017_chainsD,axiom,
! [C: set_set_a,S: set_set_a,X: set_a,Y2: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ X @ C )
=> ( ( member_set_a @ Y2 @ C )
=> ( ( ord_less_eq_set_a @ X @ Y2 )
| ( ord_less_eq_set_a @ Y2 @ X ) ) ) ) ) ).
% chainsD
thf(fact_1018_chainsD,axiom,
! [C: set_set_set_a,S: set_set_set_a,X: set_set_a,Y2: set_set_a] :
( ( member_set_set_set_a @ C @ ( chains_set_a @ S ) )
=> ( ( member_set_set_a @ X @ C )
=> ( ( member_set_set_a @ Y2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y2 )
| ( ord_le3724670747650509150_set_a @ Y2 @ X ) ) ) ) ) ).
% chainsD
thf(fact_1019_separation__t1,axiom,
! [X: a,Y2: a] :
( ( X != Y2 )
= ( ? [U4: set_a] :
( ( topolo8477419352202985285open_a @ U4 )
& ( member_a @ X @ U4 )
& ~ ( member_a @ Y2 @ U4 ) ) ) ) ).
% separation_t1
thf(fact_1020_separation__t0,axiom,
! [X: a,Y2: a] :
( ( X != Y2 )
= ( ? [U4: set_a] :
( ( topolo8477419352202985285open_a @ U4 )
& ( ( member_a @ X @ U4 )
!= ( member_a @ Y2 @ U4 ) ) ) ) ) ).
% separation_t0
thf(fact_1021_t1__space,axiom,
! [X: a,Y2: a] :
( ( X != Y2 )
=> ? [U5: set_a] :
( ( topolo8477419352202985285open_a @ U5 )
& ( member_a @ X @ U5 )
& ~ ( member_a @ Y2 @ U5 ) ) ) ).
% t1_space
thf(fact_1022_t0__space,axiom,
! [X: a,Y2: a] :
( ( X != Y2 )
=> ? [U5: set_a] :
( ( topolo8477419352202985285open_a @ U5 )
& ( ( member_a @ X @ U5 )
!= ( member_a @ Y2 @ U5 ) ) ) ) ).
% t0_space
thf(fact_1023_hausdorff,axiom,
! [X: a,Y2: a] :
( ( X != Y2 )
=> ? [U5: set_a,V4: set_a] :
( ( topolo8477419352202985285open_a @ U5 )
& ( topolo8477419352202985285open_a @ V4 )
& ( member_a @ X @ U5 )
& ( member_a @ Y2 @ V4 )
& ( ( inf_inf_set_a @ U5 @ V4 )
= bot_bot_set_a ) ) ) ).
% hausdorff
thf(fact_1024_separation__t2,axiom,
! [X: a,Y2: a] :
( ( X != Y2 )
= ( ? [U4: set_a,V5: set_a] :
( ( topolo8477419352202985285open_a @ U4 )
& ( topolo8477419352202985285open_a @ V5 )
& ( member_a @ X @ U4 )
& ( member_a @ Y2 @ V5 )
& ( ( inf_inf_set_a @ U4 @ V5 )
= bot_bot_set_a ) ) ) ) ).
% separation_t2
thf(fact_1025_mono__set,axiom,
( extend347329919781519187_set_a
= ( ^ [S4: set_set_a] :
! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( member_set_a @ X3 @ S4 )
=> ( member_set_a @ Y3 @ S4 ) ) ) ) ) ).
% mono_set
thf(fact_1026_mono__set,axiom,
( extend2808419353335425523_set_a
= ( ^ [S4: set_a] :
! [X3: a,Y3: a] :
( ( ord_less_eq_a @ X3 @ Y3 )
=> ( ( member_a @ X3 @ S4 )
=> ( member_a @ Y3 @ S4 ) ) ) ) ) ).
% mono_set
thf(fact_1027_mono__set,axiom,
( extend2184654222208438451_set_a
= ( ^ [S4: set_set_set_a] :
! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ( member_set_set_a @ X3 @ S4 )
=> ( member_set_set_a @ Y3 @ S4 ) ) ) ) ) ).
% mono_set
thf(fact_1028_open__right,axiom,
! [S: set_a,X: a,Y2: a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( member_a @ X @ S )
=> ( ( ord_less_a @ X @ Y2 )
=> ? [B3: a] :
( ( ord_less_a @ X @ B3 )
& ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ X @ B3 ) @ S ) ) ) ) ) ).
% open_right
thf(fact_1029_open__left,axiom,
! [S: set_a,X: a,Y2: a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( member_a @ X @ S )
=> ( ( ord_less_a @ Y2 @ X )
=> ? [B3: a] :
( ( ord_less_a @ B3 @ X )
& ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ B3 @ X ) @ S ) ) ) ) ) ).
% open_left
thf(fact_1030_connected__def,axiom,
( topolo2370605967727889109cted_a
= ( ^ [S4: set_a] :
~ ? [A5: set_a,B5: set_a] :
( ( topolo8477419352202985285open_a @ A5 )
& ( topolo8477419352202985285open_a @ B5 )
& ( ord_less_eq_set_a @ S4 @ ( sup_sup_set_a @ A5 @ B5 ) )
& ( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ S4 )
= bot_bot_set_a )
& ( ( inf_inf_set_a @ A5 @ S4 )
!= bot_bot_set_a )
& ( ( inf_inf_set_a @ B5 @ S4 )
!= bot_bot_set_a ) ) ) ) ).
% connected_def
thf(fact_1031_connectedI,axiom,
! [U3: set_a] :
( ! [A8: set_a] :
( ( topolo8477419352202985285open_a @ A8 )
=> ! [B8: set_a] :
( ( topolo8477419352202985285open_a @ B8 )
=> ( ( ( inf_inf_set_a @ A8 @ U3 )
!= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ B8 @ U3 )
!= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ A8 @ B8 ) @ U3 )
= bot_bot_set_a )
=> ~ ( ord_less_eq_set_a @ U3 @ ( sup_sup_set_a @ A8 @ B8 ) ) ) ) ) ) )
=> ( topolo2370605967727889109cted_a @ U3 ) ) ).
% connectedI
thf(fact_1032_less__eq__cInf__inter,axiom,
! [A: set_set_a,B: set_set_a] :
( ( condit8937546108433946286_set_a @ A )
=> ( ( condit8937546108433946286_set_a @ B )
=> ( ( ( inf_inf_set_set_a @ A @ B )
!= bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1033_less__eq__cInf__inter,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( condit1487729544555641870_set_a @ A )
=> ( ( condit1487729544555641870_set_a @ B )
=> ( ( ( inf_in1205276777018777868_set_a @ A @ B )
!= bot_bo3380559777022489994_set_a )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ ( comple9105089376463352645_set_a @ A ) @ ( comple9105089376463352645_set_a @ B ) ) @ ( comple9105089376463352645_set_a @ ( inf_in1205276777018777868_set_a @ A @ B ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1034_Compl__iff,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
= ( ~ ( member_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1035_Compl__iff,axiom,
! [C: set_a,A: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A ) )
= ( ~ ( member_set_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1036_ComplI,axiom,
! [C: a,A: set_a] :
( ~ ( member_a @ C @ A )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A ) ) ) ).
% ComplI
thf(fact_1037_ComplI,axiom,
! [C: set_a,A: set_set_a] :
( ~ ( member_set_a @ C @ A )
=> ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A ) ) ) ).
% ComplI
thf(fact_1038_compl__le__compl__iff,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y2 ) )
= ( ord_less_eq_set_a @ Y2 @ X ) ) ).
% compl_le_compl_iff
thf(fact_1039_compl__le__compl__iff,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ X ) @ ( uminus6103902357914783669_set_a @ Y2 ) )
= ( ord_le3724670747650509150_set_a @ Y2 @ X ) ) ).
% compl_le_compl_iff
thf(fact_1040_Compl__subset__Compl__iff,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) )
= ( ord_less_eq_set_a @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_1041_Compl__subset__Compl__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ A ) @ ( uminus6103902357914783669_set_a @ B ) )
= ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_1042_Compl__anti__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B ) @ ( uminus_uminus_set_a @ A ) ) ) ).
% Compl_anti_mono
thf(fact_1043_Compl__anti__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ B ) @ ( uminus6103902357914783669_set_a @ A ) ) ) ).
% Compl_anti_mono
thf(fact_1044_bdd__below_OI,axiom,
! [A: set_set_a,M2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M2 @ X2 ) )
=> ( condit8937546108433946286_set_a @ A ) ) ).
% bdd_below.I
thf(fact_1045_bdd__below_OI,axiom,
! [A: set_a,M2: a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_a @ M2 @ X2 ) )
=> ( condit5901475214736682318elow_a @ A ) ) ).
% bdd_below.I
thf(fact_1046_bdd__below_OI,axiom,
! [A: set_set_set_a,M2: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ M2 @ X2 ) )
=> ( condit1487729544555641870_set_a @ A ) ) ).
% bdd_below.I
thf(fact_1047_bdd__belowI,axiom,
! [A: set_set_a,M: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M @ X2 ) )
=> ( condit8937546108433946286_set_a @ A ) ) ).
% bdd_belowI
thf(fact_1048_bdd__belowI,axiom,
! [A: set_a,M: a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_a @ M @ X2 ) )
=> ( condit5901475214736682318elow_a @ A ) ) ).
% bdd_belowI
thf(fact_1049_bdd__belowI,axiom,
! [A: set_set_set_a,M: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ M @ X2 ) )
=> ( condit1487729544555641870_set_a @ A ) ) ).
% bdd_belowI
thf(fact_1050_bdd__below__Ici,axiom,
! [A2: a] : ( condit5901475214736682318elow_a @ ( set_ord_atLeast_a @ A2 ) ) ).
% bdd_below_Ici
thf(fact_1051_bdd__below__Ici,axiom,
! [A2: set_a] : ( condit8937546108433946286_set_a @ ( set_or8362275514725411625_set_a @ A2 ) ) ).
% bdd_below_Ici
thf(fact_1052_open__Compl,axiom,
! [S: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( topolo8477419352202985285open_a @ ( uminus_uminus_set_a @ S ) ) ) ).
% open_Compl
thf(fact_1053_closed__Compl,axiom,
! [S: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( topolo784654279908865136osed_a @ ( uminus_uminus_set_a @ S ) ) ) ).
% closed_Compl
thf(fact_1054_Compl__lessThan,axiom,
! [K: a] :
( ( uminus_uminus_set_a @ ( set_ord_lessThan_a @ K ) )
= ( set_ord_atLeast_a @ K ) ) ).
% Compl_lessThan
thf(fact_1055_Compl__atLeast,axiom,
! [K: a] :
( ( uminus_uminus_set_a @ ( set_ord_atLeast_a @ K ) )
= ( set_ord_lessThan_a @ K ) ) ).
% Compl_atLeast
thf(fact_1056_subset__Compl__singleton,axiom,
! [A: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1057_subset__Compl__singleton,axiom,
! [A: set_set_a,B2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( uminus6103902357914783669_set_a @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) )
= ( ~ ( member_set_a @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1058_subset__Compl__self__eq,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= ( A = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_1059_subset__Compl__self__eq,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( uminus6103902357914783669_set_a @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_1060_open__closed,axiom,
( topolo8477419352202985285open_a
= ( ^ [S4: set_a] : ( topolo784654279908865136osed_a @ ( uminus_uminus_set_a @ S4 ) ) ) ) ).
% open_closed
thf(fact_1061_closed__open,axiom,
( topolo784654279908865136osed_a
= ( ^ [S4: set_a] : ( topolo8477419352202985285open_a @ ( uminus_uminus_set_a @ S4 ) ) ) ) ).
% closed_open
thf(fact_1062_cInf__lower,axiom,
! [X: set_a,X5: set_set_a] :
( ( member_set_a @ X @ X5 )
=> ( ( condit8937546108433946286_set_a @ X5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X5 ) @ X ) ) ) ).
% cInf_lower
thf(fact_1063_cInf__lower,axiom,
! [X: set_set_a,X5: set_set_set_a] :
( ( member_set_set_a @ X @ X5 )
=> ( ( condit1487729544555641870_set_a @ X5 )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ X5 ) @ X ) ) ) ).
% cInf_lower
thf(fact_1064_cInf__lower2,axiom,
! [X: set_a,X5: set_set_a,Y2: set_a] :
( ( member_set_a @ X @ X5 )
=> ( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( condit8937546108433946286_set_a @ X5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X5 ) @ Y2 ) ) ) ) ).
% cInf_lower2
thf(fact_1065_cInf__lower2,axiom,
! [X: set_set_a,X5: set_set_set_a,Y2: set_set_a] :
( ( member_set_set_a @ X @ X5 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ( condit1487729544555641870_set_a @ X5 )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ X5 ) @ Y2 ) ) ) ) ).
% cInf_lower2
thf(fact_1066_bdd__below__mono,axiom,
! [B: set_a,A: set_a] :
( ( condit5901475214736682318elow_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( condit5901475214736682318elow_a @ A ) ) ) ).
% bdd_below_mono
thf(fact_1067_bdd__below__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( condit8937546108433946286_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( condit8937546108433946286_set_a @ A ) ) ) ).
% bdd_below_mono
thf(fact_1068_bdd__below_Ounfold,axiom,
( condit8937546108433946286_set_a
= ( ^ [A5: set_set_a] :
? [M3: set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( ord_less_eq_set_a @ M3 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1069_bdd__below_Ounfold,axiom,
( condit5901475214736682318elow_a
= ( ^ [A5: set_a] :
? [M3: a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( ord_less_eq_a @ M3 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1070_bdd__below_Ounfold,axiom,
( condit1487729544555641870_set_a
= ( ^ [A5: set_set_set_a] :
? [M3: set_set_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A5 )
=> ( ord_le3724670747650509150_set_a @ M3 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1071_bdd__below_OE,axiom,
! [A: set_set_a] :
( ( condit8937546108433946286_set_a @ A )
=> ~ ! [M4: set_a] :
~ ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ M4 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1072_bdd__below_OE,axiom,
! [A: set_a] :
( ( condit5901475214736682318elow_a @ A )
=> ~ ! [M4: a] :
~ ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ord_less_eq_a @ M4 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1073_bdd__below_OE,axiom,
! [A: set_set_set_a] :
( ( condit1487729544555641870_set_a @ A )
=> ~ ! [M4: set_set_a] :
~ ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
=> ( ord_le3724670747650509150_set_a @ M4 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1074_compl__mono,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y2 ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% compl_mono
thf(fact_1075_compl__mono,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ Y2 ) @ ( uminus6103902357914783669_set_a @ X ) ) ) ).
% compl_mono
thf(fact_1076_compl__le__swap1,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y2 ) ) ) ).
% compl_le_swap1
thf(fact_1077_compl__le__swap1,axiom,
! [Y2: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ ( uminus6103902357914783669_set_a @ X ) )
=> ( ord_le3724670747650509150_set_a @ X @ ( uminus6103902357914783669_set_a @ Y2 ) ) ) ).
% compl_le_swap1
thf(fact_1078_compl__le__swap2,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y2 ) @ X )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y2 ) ) ).
% compl_le_swap2
thf(fact_1079_compl__le__swap2,axiom,
! [Y2: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ Y2 ) @ X )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ X ) @ Y2 ) ) ).
% compl_le_swap2
thf(fact_1080_ComplD,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
=> ~ ( member_a @ C @ A ) ) ).
% ComplD
thf(fact_1081_ComplD,axiom,
! [C: set_a,A: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A ) )
=> ~ ( member_set_a @ C @ A ) ) ).
% ComplD
thf(fact_1082_le__cInf__iff,axiom,
! [S: set_set_a,A2: set_a] :
( ( S != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ S )
=> ( ( ord_less_eq_set_a @ A2 @ ( comple6135023378680113637_set_a @ S ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ S )
=> ( ord_less_eq_set_a @ A2 @ X3 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1083_le__cInf__iff,axiom,
! [S: set_set_set_a,A2: set_set_a] :
( ( S != bot_bo3380559777022489994_set_a )
=> ( ( condit1487729544555641870_set_a @ S )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( comple9105089376463352645_set_a @ S ) )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ S )
=> ( ord_le3724670747650509150_set_a @ A2 @ X3 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1084_cInf__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( B != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ A )
=> ( ! [B3: set_a] :
( ( member_set_a @ B3 @ B )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ X4 @ B3 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ) ) ).
% cInf_mono
thf(fact_1085_cInf__mono,axiom,
! [B: set_set_set_a,A: set_set_set_a] :
( ( B != bot_bo3380559777022489994_set_a )
=> ( ( condit1487729544555641870_set_a @ A )
=> ( ! [B3: set_set_a] :
( ( member_set_set_a @ B3 @ B )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
& ( ord_le3724670747650509150_set_a @ X4 @ B3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ A ) @ ( comple9105089376463352645_set_a @ B ) ) ) ) ) ).
% cInf_mono
thf(fact_1086_inf__shunt,axiom,
! [X: set_a,Y2: set_a] :
( ( ( inf_inf_set_a @ X @ Y2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y2 ) ) ) ).
% inf_shunt
thf(fact_1087_inf__shunt,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ( inf_inf_set_set_a @ X @ Y2 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ X @ ( uminus6103902357914783669_set_a @ Y2 ) ) ) ).
% inf_shunt
thf(fact_1088_sup__shunt,axiom,
! [X: set_a,Y2: set_a] :
( ( ( sup_sup_set_a @ X @ Y2 )
= top_top_set_a )
= ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y2 ) ) ).
% sup_shunt
thf(fact_1089_sup__shunt,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ( sup_sup_set_set_a @ X @ Y2 )
= top_top_set_set_a )
= ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ X ) @ Y2 ) ) ).
% sup_shunt
thf(fact_1090_shunt1,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ Z2 )
= ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ Y2 ) @ Z2 ) ) ) ).
% shunt1
thf(fact_1091_shunt1,axiom,
! [X: set_set_a,Y2: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y2 ) @ Z2 )
= ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ Y2 ) @ Z2 ) ) ) ).
% shunt1
thf(fact_1092_shunt2,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ ( uminus_uminus_set_a @ Y2 ) ) @ Z2 )
= ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% shunt2
thf(fact_1093_shunt2,axiom,
! [X: set_set_a,Y2: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ ( uminus6103902357914783669_set_a @ Y2 ) ) @ Z2 )
= ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ Y2 @ Z2 ) ) ) ).
% shunt2
thf(fact_1094_sup__neg__inf,axiom,
! [P3: set_a,Q3: set_a,R: set_a] :
( ( ord_less_eq_set_a @ P3 @ ( sup_sup_set_a @ Q3 @ R ) )
= ( ord_less_eq_set_a @ ( inf_inf_set_a @ P3 @ ( uminus_uminus_set_a @ Q3 ) ) @ R ) ) ).
% sup_neg_inf
thf(fact_1095_sup__neg__inf,axiom,
! [P3: set_set_a,Q3: set_set_a,R: set_set_a] :
( ( ord_le3724670747650509150_set_a @ P3 @ ( sup_sup_set_set_a @ Q3 @ R ) )
= ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ P3 @ ( uminus6103902357914783669_set_a @ Q3 ) ) @ R ) ) ).
% sup_neg_inf
thf(fact_1096_disjoint__eq__subset__Compl,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1097_disjoint__eq__subset__Compl,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A @ ( uminus6103902357914783669_set_a @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1098_cInf__superset__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( A != bot_bo3380559777022489994_set_a )
=> ( ( condit1487729544555641870_set_a @ B )
=> ( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ B ) @ ( comple9105089376463352645_set_a @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1099_cInf__superset__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ B ) @ ( comple6135023378680113637_set_a @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1100_image__eqI,axiom,
! [B2: a,F: a > a,X: a,A: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1101_image__eqI,axiom,
! [B2: set_a,F: a > set_a,X: a,A: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1102_image__eqI,axiom,
! [B2: a,F: set_a > a,X: set_a,A: set_set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_a @ X @ A )
=> ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1103_image__eqI,axiom,
! [B2: set_a,F: set_a > set_a,X: set_a,A: set_set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_a @ X @ A )
=> ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1104_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1105_subset__image__iff,axiom,
! [B: set_a,F: set_a > a,A: set_set_a] :
( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A )
& ( B
= ( image_set_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1106_subset__image__iff,axiom,
! [B: set_set_a,F: a > set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1107_subset__image__iff,axiom,
! [B: set_set_a,F: set_a > set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_set_a_set_a @ F @ A ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A )
& ( B
= ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1108_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1109_subset__imageE,axiom,
! [B: set_a,F: set_a > a,A: set_set_a] :
( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
=> ~ ! [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A )
=> ( B
!= ( image_set_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1110_subset__imageE,axiom,
! [B: set_set_a,F: a > set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1111_subset__imageE,axiom,
! [B: set_set_a,F: set_a > set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_set_a_set_a @ F @ A ) )
=> ~ ! [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A )
=> ( B
!= ( image_set_a_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1112_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1113_image__subsetI,axiom,
! [A: set_set_a,F: set_a > a,B: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1114_image__subsetI,axiom,
! [A: set_a,F: a > set_a,B: set_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1115_image__subsetI,axiom,
! [A: set_set_a,F: set_a > set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1116_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_1117_image__mono,axiom,
! [A: set_a,B: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_1118_image__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ ( image_set_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_1119_image__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A ) @ ( image_set_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_1120_INF__eq,axiom,
! [A: set_a,B: set_a,G: a > set_a,F: a > set_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) )
= ( comple6135023378680113637_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1121_INF__eq,axiom,
! [A: set_a,B: set_set_a,G: set_a > set_a,F: a > set_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ B )
& ( ord_less_eq_set_a @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: set_a] :
( ( member_set_a @ J2 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) )
= ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1122_INF__eq,axiom,
! [A: set_set_a,B: set_a,G: a > set_a,F: set_a > set_a] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A ) )
= ( comple6135023378680113637_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1123_INF__eq,axiom,
! [A: set_set_a,B: set_set_a,G: set_a > set_a,F: set_a > set_a] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ B )
& ( ord_less_eq_set_a @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: set_a] :
( ( member_set_a @ J2 @ B )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A ) )
= ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1124_INF__eq,axiom,
! [A: set_a,B: set_a,G: a > set_set_a,F: a > set_set_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_le3724670747650509150_set_a @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple9105089376463352645_set_a @ ( image_a_set_set_a @ F @ A ) )
= ( comple9105089376463352645_set_a @ ( image_a_set_set_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1125_INF__eq,axiom,
! [A: set_a,B: set_set_a,G: set_a > set_set_a,F: a > set_set_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ B )
& ( ord_le3724670747650509150_set_a @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: set_a] :
( ( member_set_a @ J2 @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple9105089376463352645_set_a @ ( image_a_set_set_a @ F @ A ) )
= ( comple9105089376463352645_set_a @ ( image_4955109552351689957_set_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1126_INF__eq,axiom,
! [A: set_set_a,B: set_a,G: a > set_set_a,F: set_a > set_set_a] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_le3724670747650509150_set_a @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple9105089376463352645_set_a @ ( image_4955109552351689957_set_a @ F @ A ) )
= ( comple9105089376463352645_set_a @ ( image_a_set_set_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1127_INF__eq,axiom,
! [A: set_set_a,B: set_set_a,G: set_a > set_set_a,F: set_a > set_set_a] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ B )
& ( ord_le3724670747650509150_set_a @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: set_a] :
( ( member_set_a @ J2 @ B )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple9105089376463352645_set_a @ ( image_4955109552351689957_set_a @ F @ A ) )
= ( comple9105089376463352645_set_a @ ( image_4955109552351689957_set_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1128_rev__image__eqI,axiom,
! [X: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1129_rev__image__eqI,axiom,
! [X: a,A: set_a,B2: set_a,F: a > set_a] :
( ( member_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1130_rev__image__eqI,axiom,
! [X: set_a,A: set_set_a,B2: a,F: set_a > a] :
( ( member_set_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1131_rev__image__eqI,axiom,
! [X: set_a,A: set_set_a,B2: set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1132_imageI,axiom,
! [X: a,A: set_a,F: a > a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_1133_imageI,axiom,
! [X: a,A: set_a,F: a > set_a] :
( ( member_a @ X @ A )
=> ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_1134_imageI,axiom,
! [X: set_a,A: set_set_a,F: set_a > a] :
( ( member_set_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_set_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_1135_imageI,axiom,
! [X: set_a,A: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A )
=> ( member_set_a @ ( F @ X ) @ ( image_set_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_1136_bdd__belowI2,axiom,
! [A: set_a,M: set_a,F: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M @ ( F @ X2 ) ) )
=> ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1137_bdd__belowI2,axiom,
! [A: set_set_a,M: set_a,F: set_a > set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M @ ( F @ X2 ) ) )
=> ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1138_bdd__belowI2,axiom,
! [A: set_a,M: a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_a @ M @ ( F @ X2 ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1139_bdd__belowI2,axiom,
! [A: set_set_a,M: a,F: set_a > a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_a @ M @ ( F @ X2 ) ) )
=> ( condit5901475214736682318elow_a @ ( image_set_a_a @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1140_bdd__belowI2,axiom,
! [A: set_a,M: set_set_a,F: a > set_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ M @ ( F @ X2 ) ) )
=> ( condit1487729544555641870_set_a @ ( image_a_set_set_a @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1141_bdd__belowI2,axiom,
! [A: set_set_a,M: set_set_a,F: set_a > set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ M @ ( F @ X2 ) ) )
=> ( condit1487729544555641870_set_a @ ( image_4955109552351689957_set_a @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1142_bdd__below_OI2,axiom,
! [A: set_a,M2: set_a,F: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M2 @ ( F @ X2 ) ) )
=> ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1143_bdd__below_OI2,axiom,
! [A: set_set_a,M2: set_a,F: set_a > set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M2 @ ( F @ X2 ) ) )
=> ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1144_bdd__below_OI2,axiom,
! [A: set_a,M2: a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_a @ M2 @ ( F @ X2 ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1145_bdd__below_OI2,axiom,
! [A: set_set_a,M2: a,F: set_a > a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_a @ M2 @ ( F @ X2 ) ) )
=> ( condit5901475214736682318elow_a @ ( image_set_a_a @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1146_bdd__below_OI2,axiom,
! [A: set_a,M2: set_set_a,F: a > set_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ M2 @ ( F @ X2 ) ) )
=> ( condit1487729544555641870_set_a @ ( image_a_set_set_a @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1147_bdd__below_OI2,axiom,
! [A: set_set_a,M2: set_set_a,F: set_a > set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ M2 @ ( F @ X2 ) ) )
=> ( condit1487729544555641870_set_a @ ( image_4955109552351689957_set_a @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1148_INF__eq__iff,axiom,
! [I5: set_a,F: a > set_a,C: set_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_less_eq_set_a @ ( F @ I3 ) @ C ) )
=> ( ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ I5 ) )
= C )
= ( ! [X3: a] :
( ( member_a @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1149_INF__eq__iff,axiom,
! [I5: set_set_a,F: set_a > set_a,C: set_a] :
( ( I5 != bot_bot_set_set_a )
=> ( ! [I3: set_a] :
( ( member_set_a @ I3 @ I5 )
=> ( ord_less_eq_set_a @ ( F @ I3 ) @ C ) )
=> ( ( ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ I5 ) )
= C )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1150_INF__eq__iff,axiom,
! [I5: set_a,F: a > set_set_a,C: set_set_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ I3 ) @ C ) )
=> ( ( ( comple9105089376463352645_set_a @ ( image_a_set_set_a @ F @ I5 ) )
= C )
= ( ! [X3: a] :
( ( member_a @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1151_INF__eq__iff,axiom,
! [I5: set_set_a,F: set_a > set_set_a,C: set_set_a] :
( ( I5 != bot_bot_set_set_a )
=> ( ! [I3: set_a] :
( ( member_set_a @ I3 @ I5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ I3 ) @ C ) )
=> ( ( ( comple9105089376463352645_set_a @ ( image_4955109552351689957_set_a @ F @ I5 ) )
= C )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1152_cINF__greatest,axiom,
! [A: set_a,M: set_a,F: a > set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M @ ( F @ X2 ) ) )
=> ( ord_less_eq_set_a @ M @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1153_cINF__greatest,axiom,
! [A: set_set_a,M: set_a,F: set_a > set_a] :
( ( A != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M @ ( F @ X2 ) ) )
=> ( ord_less_eq_set_a @ M @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1154_cINF__greatest,axiom,
! [A: set_a,M: set_set_a,F: a > set_set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ M @ ( F @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ M @ ( comple9105089376463352645_set_a @ ( image_a_set_set_a @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1155_cINF__greatest,axiom,
! [A: set_set_a,M: set_set_a,F: set_a > set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ M @ ( F @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ M @ ( comple9105089376463352645_set_a @ ( image_4955109552351689957_set_a @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1156_cINF__lower2,axiom,
! [F: a > set_a,A: set_a,X: a,U: set_a] :
( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) )
=> ( ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ ( F @ X ) @ U )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_1157_cINF__lower2,axiom,
! [F: set_a > set_a,A: set_set_a,X: set_a,U: set_a] :
( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A ) )
=> ( ( member_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ ( F @ X ) @ U )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_1158_cINF__lower2,axiom,
! [F: a > set_set_a,A: set_a,X: a,U: set_set_a] :
( ( condit1487729544555641870_set_a @ ( image_a_set_set_a @ F @ A ) )
=> ( ( member_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ X ) @ U )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ ( image_a_set_set_a @ F @ A ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_1159_cINF__lower2,axiom,
! [F: set_a > set_set_a,A: set_set_a,X: set_a,U: set_set_a] :
( ( condit1487729544555641870_set_a @ ( image_4955109552351689957_set_a @ F @ A ) )
=> ( ( member_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ X ) @ U )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ ( image_4955109552351689957_set_a @ F @ A ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_1160_cINF__lower,axiom,
! [F: a > set_a,A: set_a,X: a] :
( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) )
=> ( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) ) @ ( F @ X ) ) ) ) ).
% cINF_lower
thf(fact_1161_cINF__lower,axiom,
! [F: set_a > set_a,A: set_set_a,X: set_a] :
( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ F @ A ) )
=> ( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A ) ) @ ( F @ X ) ) ) ) ).
% cINF_lower
thf(fact_1162_cINF__lower,axiom,
! [F: a > set_set_a,A: set_a,X: a] :
( ( condit1487729544555641870_set_a @ ( image_a_set_set_a @ F @ A ) )
=> ( ( member_a @ X @ A )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ ( image_a_set_set_a @ F @ A ) ) @ ( F @ X ) ) ) ) ).
% cINF_lower
thf(fact_1163_cINF__lower,axiom,
! [F: set_a > set_set_a,A: set_set_a,X: set_a] :
( ( condit1487729544555641870_set_a @ ( image_4955109552351689957_set_a @ F @ A ) )
=> ( ( member_set_a @ X @ A )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ ( image_4955109552351689957_set_a @ F @ A ) ) @ ( F @ X ) ) ) ) ).
% cINF_lower
thf(fact_1164_cINF__superset__mono,axiom,
! [A: set_a,G: a > set_a,B: set_a,F: a > set_a] :
( ( A != bot_bot_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_a_set_a @ G @ B ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ord_less_eq_set_a @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G @ B ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1165_cINF__superset__mono,axiom,
! [A: set_a,G: a > set_set_a,B: set_a,F: a > set_set_a] :
( ( A != bot_bot_set_a )
=> ( ( condit1487729544555641870_set_a @ ( image_a_set_set_a @ G @ B ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ ( image_a_set_set_a @ G @ B ) ) @ ( comple9105089376463352645_set_a @ ( image_a_set_set_a @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1166_cINF__superset__mono,axiom,
! [A: set_set_a,G: set_a > set_a,B: set_set_a,F: set_a > set_a] :
( ( A != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_set_a_set_a @ G @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ B )
=> ( ord_less_eq_set_a @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ G @ B ) ) @ ( comple6135023378680113637_set_a @ ( image_set_a_set_a @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1167_cINF__superset__mono,axiom,
! [A: set_set_a,G: set_a > set_set_a,B: set_set_a,F: set_a > set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ( condit1487729544555641870_set_a @ ( image_4955109552351689957_set_a @ G @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ ( image_4955109552351689957_set_a @ G @ B ) ) @ ( comple9105089376463352645_set_a @ ( image_4955109552351689957_set_a @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1168_all__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1169_all__subset__image,axiom,
! [F: set_a > a,A: set_set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_set_a_a @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A )
=> ( P @ ( image_set_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1170_all__subset__image,axiom,
! [F: a > set_a,A: set_a,P: set_set_a > $o] :
( ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ ( image_a_set_a @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( P @ ( image_a_set_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1171_all__subset__image,axiom,
! [F: set_a > set_a,A: set_set_a,P: set_set_a > $o] :
( ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ ( image_set_a_set_a @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A )
=> ( P @ ( image_set_a_set_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1172_compact__Int__closed,axiom,
! [S: set_a,T3: set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( topolo8439159285038550427pact_a @ ( inf_inf_set_a @ S @ T3 ) ) ) ) ).
% compact_Int_closed
thf(fact_1173_compact__attains__inf,axiom,
! [S: set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ( S != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ S )
& ! [Xa: a] :
( ( member_a @ Xa @ S )
=> ( ord_less_eq_a @ X2 @ Xa ) ) ) ) ) ).
% compact_attains_inf
thf(fact_1174_compact__attains__sup,axiom,
! [S: set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ( S != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ S )
& ! [Xa: a] :
( ( member_a @ Xa @ S )
=> ( ord_less_eq_a @ Xa @ X2 ) ) ) ) ) ).
% compact_attains_sup
thf(fact_1175_compact__imp__closed,axiom,
! [S3: set_a] :
( ( topolo8439159285038550427pact_a @ S3 )
=> ( topolo784654279908865136osed_a @ S3 ) ) ).
% compact_imp_closed
thf(fact_1176_Fpow__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A ) @ ( finite_Fpow_a @ B ) ) ) ).
% Fpow_mono
thf(fact_1177_Fpow__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le5722252365846178494_set_a @ ( finite_Fpow_set_a @ A ) @ ( finite_Fpow_set_a @ B ) ) ) ).
% Fpow_mono
thf(fact_1178_closed__Int__compact,axiom,
! [S3: set_a,T2: set_a] :
( ( topolo784654279908865136osed_a @ S3 )
=> ( ( topolo8439159285038550427pact_a @ T2 )
=> ( topolo8439159285038550427pact_a @ ( inf_inf_set_a @ S3 @ T2 ) ) ) ) ).
% closed_Int_compact
thf(fact_1179_compact__fip,axiom,
( topolo8439159285038550427pact_a
= ( ^ [U4: set_a] :
! [A5: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( topolo784654279908865136osed_a @ X3 ) )
=> ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
=> ( ( finite_finite_set_a @ B5 )
=> ( ( inf_inf_set_a @ U4 @ ( comple6135023378680113637_set_a @ B5 ) )
!= bot_bot_set_a ) ) )
=> ( ( inf_inf_set_a @ U4 @ ( comple6135023378680113637_set_a @ A5 ) )
!= bot_bot_set_a ) ) ) ) ) ).
% compact_fip
thf(fact_1180_compact__imp__fip,axiom,
! [S: set_a,F2: set_set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ! [T6: set_a] :
( ( member_set_a @ T6 @ F2 )
=> ( topolo784654279908865136osed_a @ T6 ) )
=> ( ! [F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ F2 )
=> ( ( inf_inf_set_a @ S @ ( comple6135023378680113637_set_a @ F3 ) )
!= bot_bot_set_a ) ) )
=> ( ( inf_inf_set_a @ S @ ( comple6135023378680113637_set_a @ F2 ) )
!= bot_bot_set_a ) ) ) ) ).
% compact_imp_fip
thf(fact_1181_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A2 @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1182_finite__has__maximal2,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ord_less_eq_a @ A2 @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1183_finite__has__maximal2,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
& ( ord_le3724670747650509150_set_a @ A2 @ X2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1184_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1185_finite__has__minimal2,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ord_less_eq_a @ X2 @ A2 )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1186_finite__has__minimal2,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
& ( ord_le3724670747650509150_set_a @ X2 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1187_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_1188_finite__subset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A ) ) ) ).
% finite_subset
thf(fact_1189_infinite__super,axiom,
! [S: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S @ T3 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T3 ) ) ) ).
% infinite_super
thf(fact_1190_infinite__super,axiom,
! [S: set_set_a,T3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T3 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T3 ) ) ) ).
% infinite_super
thf(fact_1191_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_1192_rev__finite__subset,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( finite_finite_set_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_1193_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1194_finite__has__maximal,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1195_finite__has__maximal,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1196_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1197_finite__has__minimal,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1198_finite__has__minimal,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1199_finite__indexed__bound,axiom,
! [A: set_a,P: a > a > $o] :
( ( finite_finite_a @ A )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [X_1: a] : ( P @ X2 @ X_1 ) )
=> ? [M5: a] :
! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [K3: a] :
( ( ord_less_eq_a @ K3 @ M5 )
& ( P @ X4 @ K3 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1200_finite__indexed__bound,axiom,
! [A: set_set_a,P: set_a > a > $o] :
( ( finite_finite_set_a @ A )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ? [X_1: a] : ( P @ X2 @ X_1 ) )
=> ? [M5: a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ? [K3: a] :
( ( ord_less_eq_a @ K3 @ M5 )
& ( P @ X4 @ K3 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1201_cInf__le__finite,axiom,
! [X5: set_set_a,X: set_a] :
( ( finite_finite_set_a @ X5 )
=> ( ( member_set_a @ X @ X5 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X5 ) @ X ) ) ) ).
% cInf_le_finite
thf(fact_1202_cInf__le__finite,axiom,
! [X5: set_set_set_a,X: set_set_a] :
( ( finite7209287970140883943_set_a @ X5 )
=> ( ( member_set_set_a @ X @ X5 )
=> ( ord_le3724670747650509150_set_a @ ( comple9105089376463352645_set_a @ X5 ) @ X ) ) ) ).
% cInf_le_finite
thf(fact_1203_finite__imp__closed,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( topolo784654279908865136osed_a @ S ) ) ).
% finite_imp_closed
thf(fact_1204_finite__subset__image,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1205_finite__subset__image,axiom,
! [B: set_a,F: set_a > a,A: set_set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
=> ? [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A )
& ( finite_finite_set_a @ C4 )
& ( B
= ( image_set_a_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1206_finite__subset__image,axiom,
! [B: set_set_a,F: a > set_a,A: set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_set_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1207_finite__subset__image,axiom,
! [B: set_set_a,F: set_a > set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( image_set_a_set_a @ F @ A ) )
=> ? [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A )
& ( finite_finite_set_a @ C4 )
& ( B
= ( image_set_a_set_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1208_ex__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A )
& ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1209_ex__finite__subset__image,axiom,
! [F: set_a > a,A: set_set_a,P: set_a > $o] :
( ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_set_a_a @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_set_a] :
( ( finite_finite_set_a @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A )
& ( P @ ( image_set_a_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1210_ex__finite__subset__image,axiom,
! [F: a > set_a,A: set_a,P: set_set_a > $o] :
( ( ? [B5: set_set_a] :
( ( finite_finite_set_a @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ ( image_a_set_a @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A )
& ( P @ ( image_a_set_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1211_ex__finite__subset__image,axiom,
! [F: set_a > set_a,A: set_set_a,P: set_set_a > $o] :
( ( ? [B5: set_set_a] :
( ( finite_finite_set_a @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ ( image_set_a_set_a @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_set_a] :
( ( finite_finite_set_a @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A )
& ( P @ ( image_set_a_set_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1212_all__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A ) )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1213_all__finite__subset__image,axiom,
! [F: set_a > a,A: set_set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_set_a_a @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_set_a] :
( ( ( finite_finite_set_a @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A ) )
=> ( P @ ( image_set_a_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1214_all__finite__subset__image,axiom,
! [F: a > set_a,A: set_a,P: set_set_a > $o] :
( ( ! [B5: set_set_a] :
( ( ( finite_finite_set_a @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ ( image_a_set_a @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A ) )
=> ( P @ ( image_a_set_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1215_all__finite__subset__image,axiom,
! [F: set_a > set_a,A: set_set_a,P: set_set_a > $o] :
( ( ! [B5: set_set_a] :
( ( ( finite_finite_set_a @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ ( image_set_a_set_a @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_set_a] :
( ( ( finite_finite_set_a @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A ) )
=> ( P @ ( image_set_a_set_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1216_finite__subset__induct,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A3 @ A )
=> ( ~ ( member_a @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A3 @ F4 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1217_finite__subset__induct,axiom,
! [F2: set_set_a,A: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F4: set_set_a] :
( ( finite_finite_set_a @ F4 )
=> ( ( member_set_a @ A3 @ A )
=> ( ~ ( member_set_a @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_a @ A3 @ F4 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1218_finite__subset__induct_H,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A3 @ A )
=> ( ( ord_less_eq_set_a @ F4 @ A )
=> ( ~ ( member_a @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A3 @ F4 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1219_finite__subset__induct_H,axiom,
! [F2: set_set_a,A: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F4: set_set_a] :
( ( finite_finite_set_a @ F4 )
=> ( ( member_set_a @ A3 @ A )
=> ( ( ord_le3724670747650509150_set_a @ F4 @ A )
=> ( ~ ( member_set_a @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_a @ A3 @ F4 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1220_finite__Inf__in,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y4: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( member_set_a @ Y4 @ A )
=> ( member_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ A ) ) )
=> ( member_set_a @ ( comple6135023378680113637_set_a @ A ) @ A ) ) ) ) ).
% finite_Inf_in
thf(fact_1221_finite__linorder__min__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [B3: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A8 )
=> ( ord_less_a @ B3 @ X4 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_a @ B3 @ A8 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1222_ex__min__if__finite,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ S )
& ~ ? [Xa: a] :
( ( member_a @ Xa @ S )
& ( ord_less_a @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1223_infinite__growing,axiom,
! [X5: set_a] :
( ( X5 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ X5 )
=> ? [Xa: a] :
( ( member_a @ Xa @ X5 )
& ( ord_less_a @ X2 @ Xa ) ) )
=> ~ ( finite_finite_a @ X5 ) ) ) ).
% infinite_growing
thf(fact_1224_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > a] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S5: set_a] :
( ( finite_finite_a @ S5 )
=> ( ! [Y: a] :
( ( member_a @ Y @ S5 )
=> ( ord_less_eq_a @ ( F @ Y ) @ ( F @ X2 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_a @ X2 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1225_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F: set_a > a] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,S5: set_set_a] :
( ( finite_finite_set_a @ S5 )
=> ( ! [Y: set_a] :
( ( member_set_a @ Y @ S5 )
=> ( ord_less_eq_a @ ( F @ Y ) @ ( F @ X2 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_set_a @ X2 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1226_finite__linorder__max__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [B3: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A8 )
=> ( ord_less_a @ X4 @ B3 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_a @ B3 @ A8 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1227_arg__min__least,axiom,
! [S: set_a,Y2: a,F: a > a] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y2 @ S )
=> ( ord_less_eq_a @ ( F @ ( lattic3288624042836100505on_a_a @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_1228_arg__min__least,axiom,
! [S: set_set_a,Y2: set_a,F: set_a > a] :
( ( finite_finite_set_a @ S )
=> ( ( S != bot_bot_set_set_a )
=> ( ( member_set_a @ Y2 @ S )
=> ( ord_less_eq_a @ ( F @ ( lattic2642421085827009593et_a_a @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_1229_closure__closed,axiom,
! [S: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( elementary_closure_a @ S )
= S ) ) ).
% closure_closed
thf(fact_1230_closed__closure,axiom,
! [S: set_a] : ( topolo784654279908865136osed_a @ ( elementary_closure_a @ S ) ) ).
% closed_closure
thf(fact_1231_closure__subset,axiom,
! [S: set_a] : ( ord_less_eq_set_a @ S @ ( elementary_closure_a @ S ) ) ).
% closure_subset
thf(fact_1232_closure__mono,axiom,
! [S: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S @ T3 )
=> ( ord_less_eq_set_a @ ( elementary_closure_a @ S ) @ ( elementary_closure_a @ T3 ) ) ) ).
% closure_mono
thf(fact_1233_closure__unique,axiom,
! [S: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S @ T3 )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( ! [T7: set_a] :
( ( ord_less_eq_set_a @ S @ T7 )
=> ( ( topolo784654279908865136osed_a @ T7 )
=> ( ord_less_eq_set_a @ T3 @ T7 ) ) )
=> ( ( elementary_closure_a @ S )
= T3 ) ) ) ) ).
% closure_unique
thf(fact_1234_closure__minimal,axiom,
! [S: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S @ T3 )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( ord_less_eq_set_a @ ( elementary_closure_a @ S ) @ T3 ) ) ) ).
% closure_minimal
thf(fact_1235_closure__subset__eq,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ ( elementary_closure_a @ S ) @ S )
= ( topolo784654279908865136osed_a @ S ) ) ).
% closure_subset_eq
thf(fact_1236_closure__eq,axiom,
! [S: set_a] :
( ( ( elementary_closure_a @ S )
= S )
= ( topolo784654279908865136osed_a @ S ) ) ).
% closure_eq
thf(fact_1237_closure__open__Int__superset,axiom,
! [S: set_a,T3: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( elementary_closure_a @ T3 ) )
=> ( ( elementary_closure_a @ ( inf_inf_set_a @ S @ T3 ) )
= ( elementary_closure_a @ S ) ) ) ) ).
% closure_open_Int_superset
thf(fact_1238_open__Int__closure__subset,axiom,
! [S: set_a,T3: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ S @ ( elementary_closure_a @ T3 ) ) @ ( elementary_closure_a @ ( inf_inf_set_a @ S @ T3 ) ) ) ) ).
% open_Int_closure_subset
thf(fact_1239_closure__iff__nhds__not__empty,axiom,
! [X: a,X5: set_a] :
( ( member_a @ X @ ( elementary_closure_a @ X5 ) )
= ( ! [A5: set_a,S4: set_a] :
( ( ord_less_eq_set_a @ S4 @ A5 )
=> ( ( topolo8477419352202985285open_a @ S4 )
=> ( ( member_a @ X @ S4 )
=> ( ( inf_inf_set_a @ X5 @ A5 )
!= bot_bot_set_a ) ) ) ) ) ) ).
% closure_iff_nhds_not_empty
thf(fact_1240_connected__intermediate__closure,axiom,
! [S3: set_a,T2: set_a] :
( ( topolo2370605967727889109cted_a @ S3 )
=> ( ( ord_less_eq_set_a @ S3 @ T2 )
=> ( ( ord_less_eq_set_a @ T2 @ ( elementary_closure_a @ S3 ) )
=> ( topolo2370605967727889109cted_a @ T2 ) ) ) ) ).
% connected_intermediate_closure
thf(fact_1241_inf__Sup__absorb,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A2 @ ( lattic2918178356826803221_set_a @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_1242_Sup__fin_OcoboundedI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ord_less_eq_set_a @ A2 @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1243_Sup__fin_OcoboundedI,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( lattic338143333561554293_set_a @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1244_Sup__fin_Oin__idem,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X @ A )
=> ( ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A ) )
= ( lattic2918178356826803221_set_a @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1245_Sup__fin_Obounded__iff,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1246_Sup__fin_Obounded__iff,axiom,
! [A: set_set_set_a,X: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A ) @ X )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
=> ( ord_le3724670747650509150_set_a @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1247_Sup__fin_OboundedI,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A )
=> ( ord_less_eq_set_a @ A3 @ X ) )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1248_Sup__fin_OboundedI,axiom,
! [A: set_set_set_a,X: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ! [A3: set_set_a] :
( ( member_set_set_a @ A3 @ A )
=> ( ord_le3724670747650509150_set_a @ A3 @ X ) )
=> ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1249_Sup__fin_OboundedE,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X )
=> ! [A9: set_a] :
( ( member_set_a @ A9 @ A )
=> ( ord_less_eq_set_a @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1250_Sup__fin_OboundedE,axiom,
! [A: set_set_set_a,X: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A ) @ X )
=> ! [A9: set_set_a] :
( ( member_set_set_a @ A9 @ A )
=> ( ord_le3724670747650509150_set_a @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1251_Sup__fin_Osubset__imp,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( finite7209287970140883943_set_a @ B )
=> ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A ) @ ( lattic338143333561554293_set_a @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1252_Sup__fin_Osubset__imp,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1253_Sup__fin_Osubset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( B != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ B ) @ ( lattic2918178356826803221_set_a @ A ) )
= ( lattic2918178356826803221_set_a @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1254_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ~ ( member_set_a @ X @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A ) )
= ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1255_Sup__fin_Oclosed,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y4: set_a] : ( member_set_a @ ( sup_sup_set_a @ X2 @ Y4 ) @ ( insert_set_a @ X2 @ ( insert_set_a @ Y4 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic2918178356826803221_set_a @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1256_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1257_DiffI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1258_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1259_Diff__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ~ ( member_set_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1260_Diff__insert0,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1261_Diff__insert0,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A )
=> ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ B ) )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1262_insert__Diff1,axiom,
! [X: a,B: set_a,A: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1263_insert__Diff1,axiom,
! [X: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ X @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1264_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1265_Diff__eq__empty__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1266_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_1267_open__Diff,axiom,
! [S: set_a,T3: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( topolo8477419352202985285open_a @ ( minus_minus_set_a @ S @ T3 ) ) ) ) ).
% open_Diff
thf(fact_1268_closed__Diff,axiom,
! [S: set_a,T3: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo8477419352202985285open_a @ T3 )
=> ( topolo784654279908865136osed_a @ ( minus_minus_set_a @ S @ T3 ) ) ) ) ).
% closed_Diff
thf(fact_1269_sup__Inf__absorb,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ( sup_sup_set_a @ ( lattic8209813465164889211_set_a @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_1270_in__image__insert__iff,axiom,
! [B: set_set_set_a,X: set_a,A: set_set_a] :
( ! [C4: set_set_a] :
( ( member_set_set_a @ C4 @ B )
=> ~ ( member_set_a @ X @ C4 ) )
=> ( ( member_set_set_a @ A @ ( image_1042221919965026181_set_a @ ( insert_set_a @ X ) @ B ) )
= ( ( member_set_a @ X @ A )
& ( member_set_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1271_in__image__insert__iff,axiom,
! [B: set_set_a,X: a,A: set_a] :
( ! [C4: set_a] :
( ( member_set_a @ C4 @ B )
=> ~ ( member_a @ X @ C4 ) )
=> ( ( member_set_a @ A @ ( image_set_a_set_a @ ( insert_a @ X ) @ B ) )
= ( ( member_a @ X @ A )
& ( member_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1272_diff__shunt__var,axiom,
! [X: set_a,Y2: set_a] :
( ( ( minus_minus_set_a @ X @ Y2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1273_diff__shunt__var,axiom,
! [X: set_set_a,Y2: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ X @ Y2 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ X @ Y2 ) ) ).
% diff_shunt_var
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_set_a @ ( set_ord_atLeast_a @ x ) @ i ).
%------------------------------------------------------------------------------