TPTP Problem File: SLH0612^1.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 : Youngs_Inequality/0000_Youngs/prob_00179_007213__12936212_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1434 ( 456 unt; 159 typ; 0 def)
% Number of atoms : 4719 ( 960 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 12602 ( 377 ~; 133 |; 305 &;9579 @)
% ( 0 <=>;2208 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 1105 (1105 >; 0 *; 0 +; 0 <<)
% Number of symbols : 149 ( 148 usr; 6 con; 0-4 aty)
% Number of variables : 4182 ( 341 ^;3727 !; 114 ?;4182 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:27:14.467
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__b_J_J_J,type,
set_set_set_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_set_b: $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_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (148)
thf(sy_c_Abstract__Topology__2_Oretraction_001t__Nat__Onat,type,
abstra7171991951520340845on_nat: set_nat > set_nat > ( nat > nat ) > $o ).
thf(sy_c_Abstract__Topology__2_Oretraction_001tf__a,type,
abstra5157962118735104161tion_a: set_a > set_a > ( a > a ) > $o ).
thf(sy_c_Abstract__Topology__2_Oretraction_001tf__b,type,
abstra5157962118735104162tion_b: set_b > set_b > ( b > b ) > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Set__Oset_Itf__b_J,type,
monoto723715500276691686_set_b: set_nat > ( nat > nat > $o ) > ( set_b > set_b > $o ) > ( nat > set_b ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001tf__a,type,
monotone_on_nat_a: set_nat > ( nat > nat > $o ) > ( a > a > $o ) > ( nat > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001tf__b,type,
monotone_on_nat_b: set_nat > ( nat > nat > $o ) > ( b > b > $o ) > ( nat > b ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Nat__Onat_J_001tf__a,type,
monoto2395835772568396751_nat_a: set_set_nat > ( set_nat > set_nat > $o ) > ( a > a > $o ) > ( set_nat > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__b_J_001tf__a,type,
monotone_on_set_b_a: set_set_b > ( set_b > set_b > $o ) > ( a > a > $o ) > ( set_b > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001t__Nat__Onat,type,
monotone_on_a_nat: set_a > ( a > a > $o ) > ( nat > nat > $o ) > ( a > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001t__Set__Oset_Itf__b_J,type,
monotone_on_a_set_b: set_a > ( a > a > $o ) > ( set_b > set_b > $o ) > ( a > set_b ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001tf__a,type,
monotone_on_a_a: set_a > ( a > a > $o ) > ( a > a > $o ) > ( a > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001tf__b,type,
monotone_on_a_b: set_a > ( a > a > $o ) > ( b > b > $o ) > ( a > b ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__b_001t__Nat__Onat,type,
monotone_on_b_nat: set_b > ( b > b > $o ) > ( nat > nat > $o ) > ( b > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__b_001t__Set__Oset_Itf__b_J,type,
monotone_on_b_set_b: set_b > ( b > b > $o ) > ( set_b > set_b > $o ) > ( b > set_b ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__b_001tf__a,type,
monotone_on_b_a: set_b > ( b > b > $o ) > ( a > a > $o ) > ( b > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__b_001tf__b,type,
monotone_on_b_b: set_b > ( b > b > $o ) > ( b > b > $o ) > ( b > b ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
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_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
uminus613421341184616069et_nat: set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
uminus6174936397961129654_set_b: set_set_b > set_set_b ).
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_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__b_J,type,
uminus_uminus_set_b: set_b > set_b ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__b_J_J_J,type,
ord_le6262574818192256075_set_b: set_set_set_b > set_set_set_b > $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_It__Set__Oset_Itf__b_J_J,type,
ord_less_set_set_b: set_set_b > set_set_b > $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_001t__Set__Oset_Itf__b_J,type,
ord_less_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__b,type,
ord_less_b: b > b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__b_J_J_J,type,
ord_le3201067847557142847_set_b: set_set_set_b > set_set_set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
ord_le3795704787696855135_set_b: set_set_b > set_set_b > $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_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__b,type,
ord_less_eq_b: b > b > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
order_1279421399067128355et_nat: ( set_set_nat > $o ) > set_set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
order_3636894570195029656_set_b: ( set_set_b > $o ) > set_set_b ).
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_001t__Set__Oset_Itf__b_J,type,
order_Greatest_set_b: ( set_b > $o ) > set_b ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
order_Greatest_a: ( a > $o ) > a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__b,type,
order_Greatest_b: ( b > $o ) > b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
collect_set_b: ( set_b > $o ) > set_set_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__b,type,
image_nat_b: ( nat > b ) > set_nat > set_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001tf__b,type,
image_set_nat_b: ( set_nat > b ) > set_set_nat > set_b ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__b,type,
image_a_b: ( a > b ) > set_a > set_b ).
thf(sy_c_Set_Oimage_001tf__b_001t__Nat__Onat,type,
image_b_nat: ( b > nat ) > set_b > set_nat ).
thf(sy_c_Set_Oimage_001tf__b_001t__Set__Oset_It__Nat__Onat_J,type,
image_b_set_nat: ( b > set_nat ) > set_b > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
image_b_a: ( b > a ) > set_b > set_a ).
thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or9137876137106135879et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_or4832498528752608884_set_b: set_set_b > set_set_b > set_set_set_b ).
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_001t__Set__Oset_Itf__b_J,type,
set_or6288561114688587156_set_b: set_b > set_b > set_set_b ).
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_OatLeastAtMost_001tf__b,type,
set_or672772299803893940Most_b: b > b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or5410080298493297259et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_or7788675546676248656_set_b: set_set_b > set_set_b > set_set_set_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_Itf__a_J,type,
set_or2348907005316661231_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_Itf__b_J,type,
set_or2348907009619890032_set_b: set_b > set_b > set_set_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001tf__a,type,
set_or5139330845457685135Than_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001tf__b,type,
set_or5139330845457685136Than_b: b > b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Nat__Onat_J,type,
set_or1731685050470061051et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or1796310902737568945et_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_or3975068855832871818_set_b: set_set_b > set_set_set_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_Itf__a_J,type,
set_or8362275514725411625_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_Itf__b_J,type,
set_or8362275519028640426_set_b: set_b > set_set_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001tf__a,type,
set_ord_atLeast_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001tf__b,type,
set_ord_atLeast_b: b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or7210490968680142261et_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_or4087405750901549958_set_b: set_set_b > set_set_set_b ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__b_J,type,
set_ord_atMost_set_b: set_b > set_set_b ).
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_OatMost_001tf__b,type,
set_ord_atMost_b: b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or7074010630789208630et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or7489957300529979116et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_or6144489860636312079_set_b: set_set_b > set_set_b > set_set_set_b ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_Itf__a_J,type,
set_or2503527069484367278_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_Itf__b_J,type,
set_or2503527073787596079_set_b: set_b > set_b > set_set_b ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001tf__a,type,
set_or4472690218693186638Most_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001tf__b,type,
set_or4472690218693186639Most_b: b > b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or8625682525731655386et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_Itf__b_J,type,
set_or6017932781039335819_set_b: set_b > set_b > set_set_b ).
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_OgreaterThanLessThan_001tf__b,type,
set_or5939364468397584555Than_b: b > b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Nat__Onat_001t__Nat__Onat,type,
topolo1182047505939668768at_nat: set_nat > ( nat > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Nat__Onat_001tf__a,type,
topolo7330301844733086638_nat_a: set_nat > ( nat > a ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Nat__Onat_001tf__b,type,
topolo7330301844733086639_nat_b: set_nat > ( nat > b ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001tf__a_001t__Nat__Onat,type,
topolo3298370343554800208_a_nat: set_a > ( a > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001tf__a_001tf__a,type,
topolo2963393042455755902on_a_a: set_a > ( a > a ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001tf__a_001tf__b,type,
topolo2963393042455755903on_a_b: set_a > ( a > b ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001tf__b_001t__Nat__Onat,type,
topolo4533814672511194705_b_nat: set_b > ( b > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001tf__b_001tf__a,type,
topolo175937460483079869on_b_a: set_b > ( b > a ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001tf__b_001tf__b,type,
topolo175937460483079870on_b_b: set_b > ( b > b ) > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
member_set_set_b: set_set_b > set_set_set_b > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_b,type,
b2: a ).
thf(sy_v_f,type,
f: a > b ).
% Relevant facts (1274)
thf(fact_0_assms_I3_J,axiom,
ord_less_eq_a @ a2 @ b2 ).
% assms(3)
thf(fact_1__092_060open_062f_A_096_A_123a_O_Ob_125_A_092_060subseteq_062_A_123f_Aa_O_Of_Ab_125_092_060close_062,axiom,
ord_less_eq_set_b @ ( image_a_b @ f @ ( set_or672772299803893939Most_a @ a2 @ b2 ) ) @ ( set_or672772299803893940Most_b @ ( f @ a2 ) @ ( f @ b2 ) ) ).
% \<open>f ` {a..b} \<subseteq> {f a..f b}\<close>
thf(fact_2_f,axiom,
topolo2963393042455755903on_a_b @ ( set_or672772299803893939Most_a @ a2 @ b2 ) @ f ).
% f
thf(fact_3_atLeastatMost__subset__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A @ B ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4_atLeastatMost__subset__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat,D: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or9137876137106135879et_nat @ A @ B ) @ ( set_or9137876137106135879et_nat @ C @ D ) )
= ( ~ ( ord_le6893508408891458716et_nat @ A @ B )
| ( ( ord_le6893508408891458716et_nat @ C @ A )
& ( ord_le6893508408891458716et_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_5_atLeastatMost__subset__iff,axiom,
! [A: set_set_b,B: set_set_b,C: set_set_b,D: set_set_b] :
( ( ord_le3201067847557142847_set_b @ ( set_or4832498528752608884_set_b @ A @ B ) @ ( set_or4832498528752608884_set_b @ C @ D ) )
= ( ~ ( ord_le3795704787696855135_set_b @ A @ B )
| ( ( ord_le3795704787696855135_set_b @ C @ A )
& ( ord_le3795704787696855135_set_b @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_6_atLeastatMost__subset__iff,axiom,
! [A: set_b,B: set_b,C: set_b,D: set_b] :
( ( ord_le3795704787696855135_set_b @ ( set_or6288561114688587156_set_b @ A @ B ) @ ( set_or6288561114688587156_set_b @ C @ D ) )
= ( ~ ( ord_less_eq_set_b @ A @ B )
| ( ( ord_less_eq_set_b @ C @ A )
& ( ord_less_eq_set_b @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_7_atLeastatMost__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_8_atLeastatMost__subset__iff,axiom,
! [A: b,B: b,C: b,D: b] :
( ( ord_less_eq_set_b @ ( set_or672772299803893940Most_b @ A @ B ) @ ( set_or672772299803893940Most_b @ C @ D ) )
= ( ~ ( ord_less_eq_b @ A @ B )
| ( ( ord_less_eq_b @ C @ A )
& ( ord_less_eq_b @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_9_atLeastatMost__subset__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A @ B ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_10_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_11_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_12_Icc__eq__Icc,axiom,
! [L: set_set_nat,H: set_set_nat,L2: set_set_nat,H2: set_set_nat] :
( ( ( set_or9137876137106135879et_nat @ L @ H )
= ( set_or9137876137106135879et_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le6893508408891458716et_nat @ L @ H )
& ~ ( ord_le6893508408891458716et_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_13_Icc__eq__Icc,axiom,
! [L: set_set_b,H: set_set_b,L2: set_set_b,H2: set_set_b] :
( ( ( set_or4832498528752608884_set_b @ L @ H )
= ( set_or4832498528752608884_set_b @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le3795704787696855135_set_b @ L @ H )
& ~ ( ord_le3795704787696855135_set_b @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_14_Icc__eq__Icc,axiom,
! [L: set_b,H: set_b,L2: set_b,H2: set_b] :
( ( ( set_or6288561114688587156_set_b @ L @ H )
= ( set_or6288561114688587156_set_b @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_b @ L @ H )
& ~ ( ord_less_eq_set_b @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_15_Icc__eq__Icc,axiom,
! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L @ H )
= ( set_or4548717258645045905et_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_nat @ L @ H )
& ~ ( ord_less_eq_set_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_16_Icc__eq__Icc,axiom,
! [L: b,H: b,L2: b,H2: b] :
( ( ( set_or672772299803893940Most_b @ L @ H )
= ( set_or672772299803893940Most_b @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_b @ L @ H )
& ~ ( ord_less_eq_b @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_17_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_18_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_19_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_20_atLeastAtMost__iff,axiom,
! [I: set_set_nat,L: set_set_nat,U: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or9137876137106135879et_nat @ L @ U ) )
= ( ( ord_le6893508408891458716et_nat @ L @ I )
& ( ord_le6893508408891458716et_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_21_atLeastAtMost__iff,axiom,
! [I: set_set_b,L: set_set_b,U: set_set_b] :
( ( member_set_set_b @ I @ ( set_or4832498528752608884_set_b @ L @ U ) )
= ( ( ord_le3795704787696855135_set_b @ L @ I )
& ( ord_le3795704787696855135_set_b @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_22_atLeastAtMost__iff,axiom,
! [I: set_b,L: set_b,U: set_b] :
( ( member_set_b @ I @ ( set_or6288561114688587156_set_b @ L @ U ) )
= ( ( ord_less_eq_set_b @ L @ I )
& ( ord_less_eq_set_b @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_23_atLeastAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_24_atLeastAtMost__iff,axiom,
! [I: b,L: b,U: b] :
( ( member_b @ I @ ( set_or672772299803893940Most_b @ L @ U ) )
= ( ( ord_less_eq_b @ L @ I )
& ( ord_less_eq_b @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_25_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_26_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_27_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_28_subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_29_subsetI,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ! [X: set_b] :
( ( member_set_b @ X @ A2 )
=> ( member_set_b @ X @ B2 ) )
=> ( ord_le3795704787696855135_set_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_30_subsetI,axiom,
! [A2: set_b,B2: set_b] :
( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( member_b @ X @ B2 ) )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_31_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_32_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_33_subset__antisym,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_34_subset__antisym,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ( ( ord_le3795704787696855135_set_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_35_subset__antisym,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_36_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_37_image__eqI,axiom,
! [B: b,F: a > b,X2: a,A2: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A2 )
=> ( member_b @ B @ ( image_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_38_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_39_image__eqI,axiom,
! [B: a,F: nat > a,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_a @ B @ ( image_nat_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_40_image__eqI,axiom,
! [B: b,F: nat > b,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_b @ B @ ( image_nat_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_41_image__eqI,axiom,
! [B: nat,F: a > nat,X2: a,A2: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A2 )
=> ( member_nat @ B @ ( image_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_42_image__eqI,axiom,
! [B: a,F: a > a,X2: a,A2: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_43_image__eqI,axiom,
! [B: nat,F: b > nat,X2: b,A2: set_b] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A2 )
=> ( member_nat @ B @ ( image_b_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_44_image__eqI,axiom,
! [B: a,F: b > a,X2: b,A2: set_b] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A2 )
=> ( member_a @ B @ ( image_b_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_45_image__eqI,axiom,
! [B: b,F: b > b,X2: b,A2: set_b] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A2 )
=> ( member_b @ B @ ( image_b_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_46_image__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_47_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_48_order__refl,axiom,
! [X2: b] : ( ord_less_eq_b @ X2 @ X2 ) ).
% order_refl
thf(fact_49_order__refl,axiom,
! [X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_50_order__refl,axiom,
! [X2: set_set_b] : ( ord_le3795704787696855135_set_b @ X2 @ X2 ) ).
% order_refl
thf(fact_51_order__refl,axiom,
! [X2: set_b] : ( ord_less_eq_set_b @ X2 @ X2 ) ).
% order_refl
thf(fact_52_order__refl,axiom,
! [X2: a] : ( ord_less_eq_a @ X2 @ X2 ) ).
% order_refl
thf(fact_53_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_54_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_55_dual__order_Orefl,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).
% dual_order.refl
thf(fact_56_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_57_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_58_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_59_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_60_dual__order_Orefl,axiom,
! [A: b] : ( ord_less_eq_b @ A @ A ) ).
% dual_order.refl
thf(fact_61_dual__order_Orefl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% dual_order.refl
thf(fact_62_dual__order_Orefl,axiom,
! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ A @ A ) ).
% dual_order.refl
thf(fact_63_image__mono,axiom,
! [A2: set_b,B2: set_b,F: b > b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ ( image_b_b @ F @ B2 ) ) ) ).
% image_mono
thf(fact_64_image__mono,axiom,
! [A2: set_b,B2: set_b,F: b > nat] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_b_nat @ F @ A2 ) @ ( image_b_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_65_image__mono,axiom,
! [A2: set_b,B2: set_b,F: b > a] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_66_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > b] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_b @ ( image_nat_b @ F @ A2 ) @ ( image_nat_b @ F @ B2 ) ) ) ).
% image_mono
thf(fact_67_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_68_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > a] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A2 ) @ ( image_nat_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_69_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > b] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ ( image_a_b @ F @ B2 ) ) ) ).
% image_mono
thf(fact_70_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A2 ) @ ( image_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_71_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_72_image__mono,axiom,
! [A2: set_b,B2: set_b,F: b > set_nat] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_b_set_nat @ F @ A2 ) @ ( image_b_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_73_image__subsetI,axiom,
! [A2: set_nat,F: nat > b,B2: set_b] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_b @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_nat_b @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_74_image__subsetI,axiom,
! [A2: set_a,F: a > b,B2: set_b] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_b @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_75_image__subsetI,axiom,
! [A2: set_b,F: b > b,B2: set_b] :
( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( member_b @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_76_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_77_image__subsetI,axiom,
! [A2: set_a,F: a > nat,B2: set_nat] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_78_image__subsetI,axiom,
! [A2: set_b,F: b > nat,B2: set_nat] :
( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_b_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_79_image__subsetI,axiom,
! [A2: set_nat,F: nat > a,B2: set_a] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_a @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_80_image__subsetI,axiom,
! [A2: set_a,F: a > a,B2: set_a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_81_image__subsetI,axiom,
! [A2: set_b,F: b > a,B2: set_a] :
( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( member_a @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_82_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > b,B2: set_b] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_b @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_set_nat_b @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_83_subset__imageE,axiom,
! [B2: set_b,F: b > b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ ( image_b_b @ F @ A2 ) )
=> ~ ! [C2: set_b] :
( ( ord_less_eq_set_b @ C2 @ A2 )
=> ( B2
!= ( image_b_b @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_84_subset__imageE,axiom,
! [B2: set_b,F: nat > b,A2: set_nat] :
( ( ord_less_eq_set_b @ B2 @ ( image_nat_b @ F @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B2
!= ( image_nat_b @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_85_subset__imageE,axiom,
! [B2: set_b,F: a > b,A2: set_a] :
( ( ord_less_eq_set_b @ B2 @ ( image_a_b @ F @ A2 ) )
=> ~ ! [C2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( B2
!= ( image_a_b @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_86_subset__imageE,axiom,
! [B2: set_nat,F: b > nat,A2: set_b] :
( ( ord_less_eq_set_nat @ B2 @ ( image_b_nat @ F @ A2 ) )
=> ~ ! [C2: set_b] :
( ( ord_less_eq_set_b @ C2 @ A2 )
=> ( B2
!= ( image_b_nat @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_87_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_88_subset__imageE,axiom,
! [B2: set_nat,F: a > nat,A2: set_a] :
( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A2 ) )
=> ~ ! [C2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( B2
!= ( image_a_nat @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_89_subset__imageE,axiom,
! [B2: set_a,F: b > a,A2: set_b] :
( ( ord_less_eq_set_a @ B2 @ ( image_b_a @ F @ A2 ) )
=> ~ ! [C2: set_b] :
( ( ord_less_eq_set_b @ C2 @ A2 )
=> ( B2
!= ( image_b_a @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_90_subset__imageE,axiom,
! [B2: set_a,F: nat > a,A2: set_nat] :
( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B2
!= ( image_nat_a @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_91_subset__imageE,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( B2
!= ( image_a_a @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_92_subset__imageE,axiom,
! [B2: set_b,F: set_nat > b,A2: set_set_nat] :
( ( ord_less_eq_set_b @ B2 @ ( image_set_nat_b @ F @ A2 ) )
=> ~ ! [C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( B2
!= ( image_set_nat_b @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_93_image__subset__iff,axiom,
! [F: a > b,A2: set_a,B2: set_b] :
( ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ B2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_b @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_94_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_95_order__antisym__conv,axiom,
! [Y: set_b,X2: set_b] :
( ( ord_less_eq_set_b @ Y @ X2 )
=> ( ( ord_less_eq_set_b @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_96_order__antisym__conv,axiom,
! [Y: a,X2: a] :
( ( ord_less_eq_a @ Y @ X2 )
=> ( ( ord_less_eq_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_97_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_98_order__antisym__conv,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_99_order__antisym__conv,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_100_order__antisym__conv,axiom,
! [Y: b,X2: b] :
( ( ord_less_eq_b @ Y @ X2 )
=> ( ( ord_less_eq_b @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_101_order__antisym__conv,axiom,
! [Y: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_102_order__antisym__conv,axiom,
! [Y: set_set_b,X2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ Y @ X2 )
=> ( ( ord_le3795704787696855135_set_b @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_103_linorder__le__cases,axiom,
! [X2: a,Y: a] :
( ~ ( ord_less_eq_a @ X2 @ Y )
=> ( ord_less_eq_a @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_104_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_105_linorder__le__cases,axiom,
! [X2: b,Y: b] :
( ~ ( ord_less_eq_b @ X2 @ Y )
=> ( ord_less_eq_b @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_106_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_107_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_108_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > b,C: b] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_109_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_110_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_111_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_112_ord__le__eq__subst,axiom,
! [A: b,B: b,F: b > a,C: a] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_113_ord__le__eq__subst,axiom,
! [A: b,B: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_114_ord__le__eq__subst,axiom,
! [A: b,B: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_115_ord__le__eq__subst,axiom,
! [A: set_b,B: set_b,F: set_b > a,C: a] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_116_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_117_ord__eq__le__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_118_ord__eq__le__subst,axiom,
! [A: b,F: a > b,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_119_ord__eq__le__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_120_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_121_ord__eq__le__subst,axiom,
! [A: b,F: nat > b,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_122_ord__eq__le__subst,axiom,
! [A: a,F: b > a,B: b,C: b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_123_ord__eq__le__subst,axiom,
! [A: nat,F: b > nat,B: b,C: b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_124_ord__eq__le__subst,axiom,
! [A: b,F: b > b,B: b,C: b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_125_ord__eq__le__subst,axiom,
! [A: a,F: set_b > a,B: set_b,C: set_b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_126_linorder__linear,axiom,
! [X2: a,Y: a] :
( ( ord_less_eq_a @ X2 @ Y )
| ( ord_less_eq_a @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_127_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_128_linorder__linear,axiom,
! [X2: b,Y: b] :
( ( ord_less_eq_b @ X2 @ Y )
| ( ord_less_eq_b @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_129_order__eq__refl,axiom,
! [X2: set_b,Y: set_b] :
( ( X2 = Y )
=> ( ord_less_eq_set_b @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_130_order__eq__refl,axiom,
! [X2: a,Y: a] :
( ( X2 = Y )
=> ( ord_less_eq_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_131_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_132_order__eq__refl,axiom,
! [X2: set_nat,Y: set_nat] :
( ( X2 = Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_133_order__eq__refl,axiom,
! [X2: set_a,Y: set_a] :
( ( X2 = Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_134_order__eq__refl,axiom,
! [X2: b,Y: b] :
( ( X2 = Y )
=> ( ord_less_eq_b @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_135_order__eq__refl,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( X2 = Y )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_136_order__eq__refl,axiom,
! [X2: set_set_b,Y: set_set_b] :
( ( X2 = Y )
=> ( ord_le3795704787696855135_set_b @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_137_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_138_order__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_139_order__subst2,axiom,
! [A: a,B: a,F: a > b,C: b] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_140_order__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_141_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_142_order__subst2,axiom,
! [A: nat,B: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_143_order__subst2,axiom,
! [A: b,B: b,F: b > a,C: a] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_144_order__subst2,axiom,
! [A: b,B: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_145_order__subst2,axiom,
! [A: b,B: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_146_order__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > a,C: a] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_147_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_148_order__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_149_order__subst1,axiom,
! [A: a,F: b > a,B: b,C: b] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_150_order__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_151_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_152_order__subst1,axiom,
! [A: nat,F: b > nat,B: b,C: b] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_153_order__subst1,axiom,
! [A: b,F: a > b,B: a,C: a] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_154_order__subst1,axiom,
! [A: b,F: nat > b,B: nat,C: nat] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_155_order__subst1,axiom,
! [A: b,F: b > b,B: b,C: b] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_156_order__subst1,axiom,
! [A: set_b,F: a > set_b,B: a,C: a] :
( ( ord_less_eq_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_157_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_b,Z: set_b] : ( Y3 = Z ) )
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_158_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
& ( ord_less_eq_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_159_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_160_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_161_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_162_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: b,Z: b] : ( Y3 = Z ) )
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ A3 @ B3 )
& ( ord_less_eq_b @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_163_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_nat,Z: set_set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_164_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_b,Z: set_set_b] : ( Y3 = Z ) )
= ( ^ [A3: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
& ( ord_le3795704787696855135_set_b @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_165_antisym,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_166_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_167_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_168_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_169_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_170_antisym,axiom,
! [A: b,B: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_b @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_171_antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_172_antisym,axiom,
! [A: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_173_dual__order_Otrans,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_eq_set_b @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_174_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_175_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_176_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_177_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_178_dual__order_Otrans,axiom,
! [B: b,A: b,C: b] :
( ( ord_less_eq_b @ B @ A )
=> ( ( ord_less_eq_b @ C @ B )
=> ( ord_less_eq_b @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_179_dual__order_Otrans,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_le6893508408891458716et_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_180_dual__order_Otrans,axiom,
! [B: set_set_b,A: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( ord_le3795704787696855135_set_b @ C @ B )
=> ( ord_le3795704787696855135_set_b @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_181_dual__order_Oantisym,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_182_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_183_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_184_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_185_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_186_dual__order_Oantisym,axiom,
! [B: b,A: b] :
( ( ord_less_eq_b @ B @ A )
=> ( ( ord_less_eq_b @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_187_dual__order_Oantisym,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_188_dual__order_Oantisym,axiom,
! [B: set_set_b,A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_189_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_b,Z: set_b] : ( Y3 = Z ) )
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
& ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_190_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ B3 @ A3 )
& ( ord_less_eq_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_191_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_192_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_193_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_194_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: b,Z: b] : ( Y3 = Z ) )
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ B3 @ A3 )
& ( ord_less_eq_b @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_195_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_nat,Z: set_set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
& ( ord_le6893508408891458716et_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_196_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_b,Z: set_set_b] : ( Y3 = Z ) )
= ( ^ [A3: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B3 @ A3 )
& ( ord_le3795704787696855135_set_b @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_197_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: a,B4: a] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_198_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_199_linorder__wlog,axiom,
! [P: b > b > $o,A: b,B: b] :
( ! [A4: b,B4: b] :
( ( ord_less_eq_b @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: b,B4: b] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_200_order__trans,axiom,
! [X2: set_b,Y: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y )
=> ( ( ord_less_eq_set_b @ Y @ Z2 )
=> ( ord_less_eq_set_b @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_201_order__trans,axiom,
! [X2: a,Y: a,Z2: a] :
( ( ord_less_eq_a @ X2 @ Y )
=> ( ( ord_less_eq_a @ Y @ Z2 )
=> ( ord_less_eq_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_202_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_203_order__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_204_order__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_205_order__trans,axiom,
! [X2: b,Y: b,Z2: b] :
( ( ord_less_eq_b @ X2 @ Y )
=> ( ( ord_less_eq_b @ Y @ Z2 )
=> ( ord_less_eq_b @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_206_order__trans,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_207_order__trans,axiom,
! [X2: set_set_b,Y: set_set_b,Z2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X2 @ Y )
=> ( ( ord_le3795704787696855135_set_b @ Y @ Z2 )
=> ( ord_le3795704787696855135_set_b @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_208_order_Otrans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% order.trans
thf(fact_209_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_210_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_211_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_212_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_213_order_Otrans,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% order.trans
thf(fact_214_order_Otrans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_215_order_Otrans,axiom,
! [A: set_set_b,B: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ord_le3795704787696855135_set_b @ A @ C ) ) ) ).
% order.trans
thf(fact_216_order__antisym,axiom,
! [X2: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y )
=> ( ( ord_less_eq_set_b @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_217_order__antisym,axiom,
! [X2: a,Y: a] :
( ( ord_less_eq_a @ X2 @ Y )
=> ( ( ord_less_eq_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_218_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_219_order__antisym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_220_order__antisym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_221_order__antisym,axiom,
! [X2: b,Y: b] :
( ( ord_less_eq_b @ X2 @ Y )
=> ( ( ord_less_eq_b @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_222_order__antisym,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_223_order__antisym,axiom,
! [X2: set_set_b,Y: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X2 @ Y )
=> ( ( ord_le3795704787696855135_set_b @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_224_ord__le__eq__trans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_225_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_226_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_227_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_228_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_229_ord__le__eq__trans,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_230_ord__le__eq__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_231_ord__le__eq__trans,axiom,
! [A: set_set_b,B: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( B = C )
=> ( ord_le3795704787696855135_set_b @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_232_ord__eq__le__trans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( A = B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_233_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_234_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_235_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_236_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_237_ord__eq__le__trans,axiom,
! [A: b,B: b,C: b] :
( ( A = B )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_238_ord__eq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( A = B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_239_ord__eq__le__trans,axiom,
! [A: set_set_b,B: set_set_b,C: set_set_b] :
( ( A = B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ord_le3795704787696855135_set_b @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_240_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_b,Z: set_b] : ( Y3 = Z ) )
= ( ^ [X3: set_b,Y4: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y4 )
& ( ord_less_eq_set_b @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_241_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
& ( ord_less_eq_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_242_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_243_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_244_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_245_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: b,Z: b] : ( Y3 = Z ) )
= ( ^ [X3: b,Y4: b] :
( ( ord_less_eq_b @ X3 @ Y4 )
& ( ord_less_eq_b @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_246_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_nat,Z: set_set_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
& ( ord_le6893508408891458716et_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_247_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_b,Z: set_set_b] : ( Y3 = Z ) )
= ( ^ [X3: set_set_b,Y4: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y4 )
& ( ord_le3795704787696855135_set_b @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_248_le__cases3,axiom,
! [X2: a,Y: a,Z2: a] :
( ( ( ord_less_eq_a @ X2 @ Y )
=> ~ ( ord_less_eq_a @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_a @ Y @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_a @ X2 @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_a @ Z2 @ Y )
=> ~ ( ord_less_eq_a @ Y @ X2 ) )
=> ( ( ( ord_less_eq_a @ Y @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_a @ Z2 @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_249_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_250_le__cases3,axiom,
! [X2: b,Y: b,Z2: b] :
( ( ( ord_less_eq_b @ X2 @ Y )
=> ~ ( ord_less_eq_b @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_b @ Y @ X2 )
=> ~ ( ord_less_eq_b @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_b @ X2 @ Z2 )
=> ~ ( ord_less_eq_b @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_b @ Z2 @ Y )
=> ~ ( ord_less_eq_b @ Y @ X2 ) )
=> ( ( ( ord_less_eq_b @ Y @ Z2 )
=> ~ ( ord_less_eq_b @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_b @ Z2 @ X2 )
=> ~ ( ord_less_eq_b @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_251_nle__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_eq_a @ A @ B ) )
= ( ( ord_less_eq_a @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_252_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_253_nle__le,axiom,
! [A: b,B: b] :
( ( ~ ( ord_less_eq_b @ A @ B ) )
= ( ( ord_less_eq_b @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_254_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_255_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: a,F: nat > a] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_nat_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_256_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: b,F: nat > b] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_b @ B @ ( image_nat_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_257_rev__image__eqI,axiom,
! [X2: a,A2: set_a,B: nat,F: a > nat] :
( ( member_a @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_258_rev__image__eqI,axiom,
! [X2: a,A2: set_a,B: a,F: a > a] :
( ( member_a @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_259_rev__image__eqI,axiom,
! [X2: a,A2: set_a,B: b,F: a > b] :
( ( member_a @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_b @ B @ ( image_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_260_rev__image__eqI,axiom,
! [X2: b,A2: set_b,B: nat,F: b > nat] :
( ( member_b @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_b_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_261_rev__image__eqI,axiom,
! [X2: b,A2: set_b,B: a,F: b > a] :
( ( member_b @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_b_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_262_rev__image__eqI,axiom,
! [X2: b,A2: set_b,B: b,F: b > b] :
( ( member_b @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_b @ B @ ( image_b_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_263_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_264_ball__imageD,axiom,
! [F: a > b,A2: set_a,P: b > $o] :
( ! [X: b] :
( ( member_b @ X @ ( image_a_b @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_265_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_266_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_267_image__cong,axiom,
! [M: set_a,N: set_a,F: a > b,G: a > b] :
( ( M = N )
=> ( ! [X: a] :
( ( member_a @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_a_b @ F @ M )
= ( image_a_b @ G @ N ) ) ) ) ).
% image_cong
thf(fact_268_bex__imageD,axiom,
! [F: a > b,A2: set_a,P: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( image_a_b @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: a] :
( ( member_a @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_269_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_270_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_271_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_272_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_273_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_274_mem__Collect__eq,axiom,
! [A: set_b,P: set_b > $o] :
( ( member_set_b @ A @ ( collect_set_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_275_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_276_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_277_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_278_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_279_Collect__mem__eq,axiom,
! [A2: set_set_b] :
( ( collect_set_b
@ ^ [X3: set_b] : ( member_set_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_280_image__iff,axiom,
! [Z2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_281_image__iff,axiom,
! [Z2: b,F: a > b,A2: set_a] :
( ( member_b @ Z2 @ ( image_a_b @ F @ A2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_282_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_283_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > a] :
( ( member_nat @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( image_nat_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_284_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > b] :
( ( member_nat @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( image_nat_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_285_imageI,axiom,
! [X2: a,A2: set_a,F: a > nat] :
( ( member_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_286_imageI,axiom,
! [X2: a,A2: set_a,F: a > a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_287_imageI,axiom,
! [X2: a,A2: set_a,F: a > b] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( image_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_288_imageI,axiom,
! [X2: b,A2: set_b,F: b > nat] :
( ( member_b @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_b_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_289_imageI,axiom,
! [X2: b,A2: set_b,F: b > a] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( image_b_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_290_imageI,axiom,
! [X2: b,A2: set_b,F: b > b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( image_b_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_291_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_292_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_293_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_294_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_295_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_296_Collect__mono__iff,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) )
= ( ! [X3: set_b] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_297_set__eq__subset,axiom,
( ( ^ [Y3: set_b,Z: set_b] : ( Y3 = Z ) )
= ( ^ [A5: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ A5 @ B5 )
& ( ord_less_eq_set_b @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_298_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_299_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [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_300_set__eq__subset,axiom,
( ( ^ [Y3: set_set_nat,Z: set_set_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B5 )
& ( ord_le6893508408891458716et_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_301_set__eq__subset,axiom,
( ( ^ [Y3: set_set_b,Z: set_set_b] : ( Y3 = Z ) )
= ( ^ [A5: set_set_b,B5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A5 @ B5 )
& ( ord_le3795704787696855135_set_b @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_302_subset__trans,axiom,
! [A2: set_b,B2: set_b,C3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C3 )
=> ( ord_less_eq_set_b @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_303_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ord_less_eq_set_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_304_subset__trans,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_305_subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C3 )
=> ( ord_le6893508408891458716et_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_306_subset__trans,axiom,
! [A2: set_set_b,B2: set_set_b,C3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ( ( ord_le3795704787696855135_set_b @ B2 @ C3 )
=> ( ord_le3795704787696855135_set_b @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_307_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X: b] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_308_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_309_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_310_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_311_Collect__mono,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ! [X: set_b] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) ) ) ).
% Collect_mono
thf(fact_312_subset__refl,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ A2 ) ).
% subset_refl
thf(fact_313_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_314_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_315_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_316_subset__refl,axiom,
! [A2: set_set_b] : ( ord_le3795704787696855135_set_b @ A2 @ A2 ) ).
% subset_refl
thf(fact_317_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B5: set_b] :
! [T: b] :
( ( member_b @ T @ A5 )
=> ( member_b @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_318_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_319_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_320_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A5 )
=> ( member_set_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_321_subset__iff,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A5: set_set_b,B5: set_set_b] :
! [T: set_b] :
( ( member_set_b @ T @ A5 )
=> ( member_set_b @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_322_equalityD2,axiom,
! [A2: set_b,B2: set_b] :
( ( A2 = B2 )
=> ( ord_less_eq_set_b @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_323_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_324_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_325_equalityD2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_326_equalityD2,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( A2 = B2 )
=> ( ord_le3795704787696855135_set_b @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_327_equalityD1,axiom,
! [A2: set_b,B2: set_b] :
( ( A2 = B2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_328_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_329_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_330_equalityD1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_331_equalityD1,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( A2 = B2 )
=> ( ord_le3795704787696855135_set_b @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_332_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B5: set_b] :
! [X3: b] :
( ( member_b @ X3 @ A5 )
=> ( member_b @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_333_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_334_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_335_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A5 )
=> ( member_set_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_336_subset__eq,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A5: set_set_b,B5: set_set_b] :
! [X3: set_b] :
( ( member_set_b @ X3 @ A5 )
=> ( member_set_b @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_337_equalityE,axiom,
! [A2: set_b,B2: set_b] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ~ ( ord_less_eq_set_b @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_338_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_339_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_340_equalityE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ~ ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_341_equalityE,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( A2 = B2 )
=> ~ ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ~ ( ord_le3795704787696855135_set_b @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_342_subsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_343_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_344_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_345_subsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_346_subsetD,axiom,
! [A2: set_set_b,B2: set_set_b,C: set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ( ( member_set_b @ C @ A2 )
=> ( member_set_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_347_in__mono,axiom,
! [A2: set_b,B2: set_b,X2: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ X2 @ A2 )
=> ( member_b @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_348_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_349_in__mono,axiom,
! [A2: set_a,B2: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_350_in__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_351_in__mono,axiom,
! [A2: set_set_b,B2: set_set_b,X2: set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ( ( member_set_b @ X2 @ A2 )
=> ( member_set_b @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_352_subset__image__iff,axiom,
! [B2: set_b,F: b > b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ ( image_b_b @ F @ A2 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A2 )
& ( B2
= ( image_b_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_353_subset__image__iff,axiom,
! [B2: set_b,F: nat > b,A2: set_nat] :
( ( ord_less_eq_set_b @ B2 @ ( image_nat_b @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_354_subset__image__iff,axiom,
! [B2: set_b,F: a > b,A2: set_a] :
( ( ord_less_eq_set_b @ B2 @ ( image_a_b @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_355_subset__image__iff,axiom,
! [B2: set_nat,F: b > nat,A2: set_b] :
( ( ord_less_eq_set_nat @ B2 @ ( image_b_nat @ F @ A2 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A2 )
& ( B2
= ( image_b_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_356_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_357_subset__image__iff,axiom,
! [B2: set_nat,F: a > nat,A2: set_a] :
( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_358_subset__image__iff,axiom,
! [B2: set_a,F: b > a,A2: set_b] :
( ( ord_less_eq_set_a @ B2 @ ( image_b_a @ F @ A2 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A2 )
& ( B2
= ( image_b_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_359_subset__image__iff,axiom,
! [B2: set_a,F: nat > a,A2: set_nat] :
( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_360_subset__image__iff,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_361_subset__image__iff,axiom,
! [B2: set_b,F: set_nat > b,A2: set_set_nat] :
( ( ord_less_eq_set_b @ B2 @ ( image_set_nat_b @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_set_nat_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_362_IVT2_H,axiom,
! [F: a > a,B: a,Y: a,A: a] :
( ( ord_less_eq_a @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_a @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ( topolo2963393042455755902on_a_a @ ( set_or672772299803893939Most_a @ A @ B ) @ F )
=> ? [X: a] :
( ( ord_less_eq_a @ A @ X )
& ( ord_less_eq_a @ X @ B )
& ( ( F @ X )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_363_IVT2_H,axiom,
! [F: a > nat,B: a,Y: nat,A: a] :
( ( ord_less_eq_nat @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_nat @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ( topolo3298370343554800208_a_nat @ ( set_or672772299803893939Most_a @ A @ B ) @ F )
=> ? [X: a] :
( ( ord_less_eq_a @ A @ X )
& ( ord_less_eq_a @ X @ B )
& ( ( F @ X )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_364_IVT2_H,axiom,
! [F: a > b,B: a,Y: b,A: a] :
( ( ord_less_eq_b @ ( F @ B ) @ Y )
=> ( ( ord_less_eq_b @ Y @ ( F @ A ) )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ( topolo2963393042455755903on_a_b @ ( set_or672772299803893939Most_a @ A @ B ) @ F )
=> ? [X: a] :
( ( ord_less_eq_a @ A @ X )
& ( ord_less_eq_a @ X @ B )
& ( ( F @ X )
= Y ) ) ) ) ) ) ).
% IVT2'
thf(fact_365_IVT_H,axiom,
! [F: a > a,A: a,Y: a,B: a] :
( ( ord_less_eq_a @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_a @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ( topolo2963393042455755902on_a_a @ ( set_or672772299803893939Most_a @ A @ B ) @ F )
=> ? [X: a] :
( ( ord_less_eq_a @ A @ X )
& ( ord_less_eq_a @ X @ B )
& ( ( F @ X )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_366_IVT_H,axiom,
! [F: a > nat,A: a,Y: nat,B: a] :
( ( ord_less_eq_nat @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_nat @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ( topolo3298370343554800208_a_nat @ ( set_or672772299803893939Most_a @ A @ B ) @ F )
=> ? [X: a] :
( ( ord_less_eq_a @ A @ X )
& ( ord_less_eq_a @ X @ B )
& ( ( F @ X )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_367_IVT_H,axiom,
! [F: a > b,A: a,Y: b,B: a] :
( ( ord_less_eq_b @ ( F @ A ) @ Y )
=> ( ( ord_less_eq_b @ Y @ ( F @ B ) )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ( topolo2963393042455755903on_a_b @ ( set_or672772299803893939Most_a @ A @ B ) @ F )
=> ? [X: a] :
( ( ord_less_eq_a @ A @ X )
& ( ord_less_eq_a @ X @ B )
& ( ( F @ X )
= Y ) ) ) ) ) ) ).
% IVT'
thf(fact_368_invertible__fixpoint__property,axiom,
! [T2: set_b,I: b > b,S: set_b,R: b > b,G: b > b] :
( ( topolo175937460483079870on_b_b @ T2 @ I )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ I @ T2 ) @ S )
=> ( ( topolo175937460483079870on_b_b @ S @ R )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ R @ S ) @ T2 )
=> ( ! [Y2: b] :
( ( member_b @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: b > b] :
( ( topolo175937460483079870on_b_b @ S @ F2 )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ F2 @ S ) @ S )
=> ? [X4: b] :
( ( member_b @ X4 @ S )
& ( ( F2 @ X4 )
= X4 ) ) ) )
=> ( ( topolo175937460483079870on_b_b @ T2 @ G )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ G @ T2 ) @ T2 )
=> ~ ! [Y2: b] :
( ( member_b @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_369_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > b,S: set_b,R: b > nat,G: nat > nat] :
( ( topolo7330301844733086639_nat_b @ T2 @ I )
=> ( ( ord_less_eq_set_b @ ( image_nat_b @ I @ T2 ) @ S )
=> ( ( topolo4533814672511194705_b_nat @ S @ R )
=> ( ( ord_less_eq_set_nat @ ( image_b_nat @ R @ S ) @ T2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: b > b] :
( ( topolo175937460483079870on_b_b @ S @ F2 )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ F2 @ S ) @ S )
=> ? [X4: b] :
( ( member_b @ X4 @ S )
& ( ( F2 @ X4 )
= X4 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T2 ) @ T2 )
=> ~ ! [Y2: nat] :
( ( member_nat @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_370_invertible__fixpoint__property,axiom,
! [T2: set_b,I: b > nat,S: set_nat,R: nat > b,G: b > b] :
( ( topolo4533814672511194705_b_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_b_nat @ I @ T2 ) @ S )
=> ( ( topolo7330301844733086639_nat_b @ S @ R )
=> ( ( ord_less_eq_set_b @ ( image_nat_b @ R @ S ) @ T2 )
=> ( ! [Y2: b] :
( ( member_b @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: nat > nat] :
( ( topolo1182047505939668768at_nat @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S ) @ S )
=> ? [X4: nat] :
( ( member_nat @ X4 @ S )
& ( ( F2 @ X4 )
= X4 ) ) ) )
=> ( ( topolo175937460483079870on_b_b @ T2 @ G )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ G @ T2 ) @ T2 )
=> ~ ! [Y2: b] :
( ( member_b @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_371_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > nat,S: set_nat,R: nat > nat,G: nat > nat] :
( ( topolo1182047505939668768at_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ I @ T2 ) @ S )
=> ( ( topolo1182047505939668768at_nat @ S @ R )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ S ) @ T2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: nat > nat] :
( ( topolo1182047505939668768at_nat @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S ) @ S )
=> ? [X4: nat] :
( ( member_nat @ X4 @ S )
& ( ( F2 @ X4 )
= X4 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T2 ) @ T2 )
=> ~ ! [Y2: nat] :
( ( member_nat @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_372_invertible__fixpoint__property,axiom,
! [T2: set_a,I: a > nat,S: set_nat,R: nat > a,G: a > a] :
( ( topolo3298370343554800208_a_nat @ T2 @ I )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ I @ T2 ) @ S )
=> ( ( topolo7330301844733086638_nat_a @ S @ R )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ R @ S ) @ T2 )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: nat > nat] :
( ( topolo1182047505939668768at_nat @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S ) @ S )
=> ? [X4: nat] :
( ( member_nat @ X4 @ S )
& ( ( F2 @ X4 )
= X4 ) ) ) )
=> ( ( topolo2963393042455755902on_a_a @ T2 @ G )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G @ T2 ) @ T2 )
=> ~ ! [Y2: a] :
( ( member_a @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_373_invertible__fixpoint__property,axiom,
! [T2: set_nat,I: nat > a,S: set_a,R: a > nat,G: nat > nat] :
( ( topolo7330301844733086638_nat_a @ T2 @ I )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ I @ T2 ) @ S )
=> ( ( topolo3298370343554800208_a_nat @ S @ R )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ R @ S ) @ T2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: a > a] :
( ( topolo2963393042455755902on_a_a @ S @ F2 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ S ) @ S )
=> ? [X4: a] :
( ( member_a @ X4 @ S )
& ( ( F2 @ X4 )
= X4 ) ) ) )
=> ( ( topolo1182047505939668768at_nat @ T2 @ G )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ T2 ) @ T2 )
=> ~ ! [Y2: nat] :
( ( member_nat @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_374_invertible__fixpoint__property,axiom,
! [T2: set_a,I: a > a,S: set_a,R: a > a,G: a > a] :
( ( topolo2963393042455755902on_a_a @ T2 @ I )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ I @ T2 ) @ S )
=> ( ( topolo2963393042455755902on_a_a @ S @ R )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R @ S ) @ T2 )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: a > a] :
( ( topolo2963393042455755902on_a_a @ S @ F2 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ S ) @ S )
=> ? [X4: a] :
( ( member_a @ X4 @ S )
& ( ( F2 @ X4 )
= X4 ) ) ) )
=> ( ( topolo2963393042455755902on_a_a @ T2 @ G )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G @ T2 ) @ T2 )
=> ~ ! [Y2: a] :
( ( member_a @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_375_invertible__fixpoint__property,axiom,
! [T2: set_b,I: b > a,S: set_a,R: a > b,G: b > b] :
( ( topolo175937460483079869on_b_a @ T2 @ I )
=> ( ( ord_less_eq_set_a @ ( image_b_a @ I @ T2 ) @ S )
=> ( ( topolo2963393042455755903on_a_b @ S @ R )
=> ( ( ord_less_eq_set_b @ ( image_a_b @ R @ S ) @ T2 )
=> ( ! [Y2: b] :
( ( member_b @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: a > a] :
( ( topolo2963393042455755902on_a_a @ S @ F2 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ S ) @ S )
=> ? [X4: a] :
( ( member_a @ X4 @ S )
& ( ( F2 @ X4 )
= X4 ) ) ) )
=> ( ( topolo175937460483079870on_b_b @ T2 @ G )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ G @ T2 ) @ T2 )
=> ~ ! [Y2: b] :
( ( member_b @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_376_invertible__fixpoint__property,axiom,
! [T2: set_a,I: a > b,S: set_b,R: b > a,G: a > a] :
( ( topolo2963393042455755903on_a_b @ T2 @ I )
=> ( ( ord_less_eq_set_b @ ( image_a_b @ I @ T2 ) @ S )
=> ( ( topolo175937460483079869on_b_a @ S @ R )
=> ( ( ord_less_eq_set_a @ ( image_b_a @ R @ S ) @ T2 )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ T2 )
=> ( ( R @ ( I @ Y2 ) )
= Y2 ) )
=> ( ! [F2: b > b] :
( ( topolo175937460483079870on_b_b @ S @ F2 )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ F2 @ S ) @ S )
=> ? [X4: b] :
( ( member_b @ X4 @ S )
& ( ( F2 @ X4 )
= X4 ) ) ) )
=> ( ( topolo2963393042455755902on_a_a @ T2 @ G )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G @ T2 ) @ T2 )
=> ~ ! [Y2: a] :
( ( member_a @ Y2 @ T2 )
=> ( ( G @ Y2 )
!= Y2 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_377_continuous__on__subset,axiom,
! [S2: set_a,F: a > b,T3: set_a] :
( ( topolo2963393042455755903on_a_b @ S2 @ F )
=> ( ( ord_less_eq_set_a @ T3 @ S2 )
=> ( topolo2963393042455755903on_a_b @ T3 @ F ) ) ) ).
% continuous_on_subset
thf(fact_378_all__subset__image,axiom,
! [F: b > b,A2: set_b,P: set_b > $o] :
( ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ ( image_b_b @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ A2 )
=> ( P @ ( image_b_b @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_379_all__subset__image,axiom,
! [F: nat > b,A2: set_nat,P: set_b > $o] :
( ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ ( image_nat_b @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( P @ ( image_nat_b @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_380_all__subset__image,axiom,
! [F: a > b,A2: set_a,P: set_b > $o] :
( ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ ( image_a_b @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A2 )
=> ( P @ ( image_a_b @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_381_all__subset__image,axiom,
! [F: b > nat,A2: set_b,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_b_nat @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ A2 )
=> ( P @ ( image_b_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_382_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_383_all__subset__image,axiom,
! [F: a > nat,A2: set_a,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_a_nat @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A2 )
=> ( P @ ( image_a_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_384_all__subset__image,axiom,
! [F: b > a,A2: set_b,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_b_a @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ A2 )
=> ( P @ ( image_b_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_385_all__subset__image,axiom,
! [F: nat > a,A2: set_nat,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_nat_a @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( P @ ( image_nat_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_386_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A2 )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_387_all__subset__image,axiom,
! [F: set_nat > b,A2: set_set_nat,P: set_b > $o] :
( ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ ( image_set_nat_b @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A2 )
=> ( P @ ( image_set_nat_b @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_388_assms_I1_J,axiom,
monotone_on_a_b @ ( set_or672772299803893939Most_a @ a2 @ b2 ) @ ord_less_a @ ord_less_b @ f ).
% assms(1)
thf(fact_389_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_390_Icc__subset__Ici__iff,axiom,
! [L: set_set_nat,H: set_set_nat,L2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or9137876137106135879et_nat @ L @ H ) @ ( set_or1796310902737568945et_nat @ L2 ) )
= ( ~ ( ord_le6893508408891458716et_nat @ L @ H )
| ( ord_le6893508408891458716et_nat @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_391_Icc__subset__Ici__iff,axiom,
! [L: set_set_b,H: set_set_b,L2: set_set_b] :
( ( ord_le3201067847557142847_set_b @ ( set_or4832498528752608884_set_b @ L @ H ) @ ( set_or3975068855832871818_set_b @ L2 ) )
= ( ~ ( ord_le3795704787696855135_set_b @ L @ H )
| ( ord_le3795704787696855135_set_b @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_392_Icc__subset__Ici__iff,axiom,
! [L: b,H: b,L2: b] :
( ( ord_less_eq_set_b @ ( set_or672772299803893940Most_b @ L @ H ) @ ( set_ord_atLeast_b @ L2 ) )
= ( ~ ( ord_less_eq_b @ L @ H )
| ( ord_less_eq_b @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_393_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_394_Icc__subset__Ici__iff,axiom,
! [L: nat,H: nat,L2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atLeast_nat @ L2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_395_Icc__subset__Ici__iff,axiom,
! [L: set_nat,H: set_nat,L2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H ) @ ( set_or1731685050470061051et_nat @ L2 ) )
= ( ~ ( ord_less_eq_set_nat @ L @ H )
| ( ord_less_eq_set_nat @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_396_Icc__subset__Ici__iff,axiom,
! [L: set_b,H: set_b,L2: set_b] :
( ( ord_le3795704787696855135_set_b @ ( set_or6288561114688587156_set_b @ L @ H ) @ ( set_or8362275519028640426_set_b @ L2 ) )
= ( ~ ( ord_less_eq_set_b @ L @ H )
| ( ord_less_eq_set_b @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_397_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_398_Icc__subset__Iic__iff,axiom,
! [L: set_set_nat,H: set_set_nat,H2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or9137876137106135879et_nat @ L @ H ) @ ( set_or7210490968680142261et_nat @ H2 ) )
= ( ~ ( ord_le6893508408891458716et_nat @ L @ H )
| ( ord_le6893508408891458716et_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_399_Icc__subset__Iic__iff,axiom,
! [L: set_set_b,H: set_set_b,H2: set_set_b] :
( ( ord_le3201067847557142847_set_b @ ( set_or4832498528752608884_set_b @ L @ H ) @ ( set_or4087405750901549958_set_b @ H2 ) )
= ( ~ ( ord_le3795704787696855135_set_b @ L @ H )
| ( ord_le3795704787696855135_set_b @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_400_Icc__subset__Iic__iff,axiom,
! [L: b,H: b,H2: b] :
( ( ord_less_eq_set_b @ ( set_or672772299803893940Most_b @ L @ H ) @ ( set_ord_atMost_b @ H2 ) )
= ( ~ ( ord_less_eq_b @ L @ H )
| ( ord_less_eq_b @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_401_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_402_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_403_Icc__subset__Iic__iff,axiom,
! [L: set_nat,H: set_nat,H2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H ) @ ( set_or4236626031148496127et_nat @ H2 ) )
= ( ~ ( ord_less_eq_set_nat @ L @ H )
| ( ord_less_eq_set_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_404_Icc__subset__Iic__iff,axiom,
! [L: set_b,H: set_b,H2: set_b] :
( ( ord_le3795704787696855135_set_b @ ( set_or6288561114688587156_set_b @ L @ H ) @ ( set_ord_atMost_set_b @ H2 ) )
= ( ~ ( ord_less_eq_set_b @ L @ H )
| ( ord_less_eq_set_b @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_405_Greatest__equality,axiom,
! [P: set_b > $o,X2: set_b] :
( ( P @ X2 )
=> ( ! [Y2: set_b] :
( ( P @ Y2 )
=> ( ord_less_eq_set_b @ Y2 @ X2 ) )
=> ( ( order_Greatest_set_b @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_406_Greatest__equality,axiom,
! [P: a > $o,X2: a] :
( ( P @ X2 )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_407_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_408_Greatest__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_409_Greatest__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_410_Greatest__equality,axiom,
! [P: b > $o,X2: b] :
( ( P @ X2 )
=> ( ! [Y2: b] :
( ( P @ Y2 )
=> ( ord_less_eq_b @ Y2 @ X2 ) )
=> ( ( order_Greatest_b @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_411_Greatest__equality,axiom,
! [P: set_set_nat > $o,X2: set_set_nat] :
( ( P @ X2 )
=> ( ! [Y2: set_set_nat] :
( ( P @ Y2 )
=> ( ord_le6893508408891458716et_nat @ Y2 @ X2 ) )
=> ( ( order_1279421399067128355et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_412_Greatest__equality,axiom,
! [P: set_set_b > $o,X2: set_set_b] :
( ( P @ X2 )
=> ( ! [Y2: set_set_b] :
( ( P @ Y2 )
=> ( ord_le3795704787696855135_set_b @ Y2 @ X2 ) )
=> ( ( order_3636894570195029656_set_b @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_413_GreatestI2__order,axiom,
! [P: set_b > $o,X2: set_b,Q: set_b > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_b] :
( ( P @ Y2 )
=> ( ord_less_eq_set_b @ Y2 @ X2 ) )
=> ( ! [X: set_b] :
( ( P @ X )
=> ( ! [Y5: set_b] :
( ( P @ Y5 )
=> ( ord_less_eq_set_b @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_414_GreatestI2__order,axiom,
! [P: a > $o,X2: a,Q: a > $o] :
( ( P @ X2 )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ! [X: a] :
( ( P @ X )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_415_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_416_GreatestI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ! [X: set_nat] :
( ( P @ X )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_417_GreatestI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_418_GreatestI2__order,axiom,
! [P: b > $o,X2: b,Q: b > $o] :
( ( P @ X2 )
=> ( ! [Y2: b] :
( ( P @ Y2 )
=> ( ord_less_eq_b @ Y2 @ X2 ) )
=> ( ! [X: b] :
( ( P @ X )
=> ( ! [Y5: b] :
( ( P @ Y5 )
=> ( ord_less_eq_b @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_419_GreatestI2__order,axiom,
! [P: set_set_nat > $o,X2: set_set_nat,Q: set_set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_set_nat] :
( ( P @ Y2 )
=> ( ord_le6893508408891458716et_nat @ Y2 @ X2 ) )
=> ( ! [X: set_set_nat] :
( ( P @ X )
=> ( ! [Y5: set_set_nat] :
( ( P @ Y5 )
=> ( ord_le6893508408891458716et_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_1279421399067128355et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_420_GreatestI2__order,axiom,
! [P: set_set_b > $o,X2: set_set_b,Q: set_set_b > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_set_b] :
( ( P @ Y2 )
=> ( ord_le3795704787696855135_set_b @ Y2 @ X2 ) )
=> ( ! [X: set_set_b] :
( ( P @ X )
=> ( ! [Y5: set_set_b] :
( ( P @ Y5 )
=> ( ord_le3795704787696855135_set_b @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_3636894570195029656_set_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_421_continuous__on__eq,axiom,
! [S2: set_a,F: a > b,G: a > b] :
( ( topolo2963393042455755903on_a_b @ S2 @ F )
=> ( ! [X: a] :
( ( member_a @ X @ S2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( topolo2963393042455755903on_a_b @ S2 @ G ) ) ) ).
% continuous_on_eq
thf(fact_422_continuous__on__cong,axiom,
! [S2: set_a,T3: set_a,F: a > b,G: a > b] :
( ( S2 = T3 )
=> ( ! [X: a] :
( ( member_a @ X @ T3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( topolo2963393042455755903on_a_b @ S2 @ F )
= ( topolo2963393042455755903on_a_b @ T3 @ G ) ) ) ) ).
% continuous_on_cong
thf(fact_423_atMost__eq__iff,axiom,
! [X2: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X2 )
= ( set_ord_atMost_nat @ Y ) )
= ( X2 = Y ) ) ).
% atMost_eq_iff
thf(fact_424_atLeast__eq__iff,axiom,
! [X2: nat,Y: nat] :
( ( ( set_ord_atLeast_nat @ X2 )
= ( set_ord_atLeast_nat @ Y ) )
= ( X2 = Y ) ) ).
% atLeast_eq_iff
thf(fact_425_atMost__iff,axiom,
! [I: set_b,K: set_b] :
( ( member_set_b @ I @ ( set_ord_atMost_set_b @ K ) )
= ( ord_less_eq_set_b @ I @ K ) ) ).
% atMost_iff
thf(fact_426_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_427_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_428_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_429_atMost__iff,axiom,
! [I: b,K: b] :
( ( member_b @ I @ ( set_ord_atMost_b @ K ) )
= ( ord_less_eq_b @ I @ K ) ) ).
% atMost_iff
thf(fact_430_atMost__iff,axiom,
! [I: set_set_nat,K: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or7210490968680142261et_nat @ K ) )
= ( ord_le6893508408891458716et_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_431_atMost__iff,axiom,
! [I: set_set_b,K: set_set_b] :
( ( member_set_set_b @ I @ ( set_or4087405750901549958_set_b @ K ) )
= ( ord_le3795704787696855135_set_b @ I @ K ) ) ).
% atMost_iff
thf(fact_432_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_433_atLeast__iff,axiom,
! [I: set_b,K: set_b] :
( ( member_set_b @ I @ ( set_or8362275519028640426_set_b @ K ) )
= ( ord_less_eq_set_b @ K @ I ) ) ).
% atLeast_iff
thf(fact_434_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_435_atLeast__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or1731685050470061051et_nat @ K ) )
= ( ord_less_eq_set_nat @ K @ I ) ) ).
% atLeast_iff
thf(fact_436_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_437_atLeast__iff,axiom,
! [I: b,K: b] :
( ( member_b @ I @ ( set_ord_atLeast_b @ K ) )
= ( ord_less_eq_b @ K @ I ) ) ).
% atLeast_iff
thf(fact_438_atLeast__iff,axiom,
! [I: set_set_nat,K: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or1796310902737568945et_nat @ K ) )
= ( ord_le6893508408891458716et_nat @ K @ I ) ) ).
% atLeast_iff
thf(fact_439_atLeast__iff,axiom,
! [I: set_set_b,K: set_set_b] :
( ( member_set_set_b @ I @ ( set_or3975068855832871818_set_b @ K ) )
= ( ord_le3795704787696855135_set_b @ K @ I ) ) ).
% atLeast_iff
thf(fact_440_atLeast__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atLeast_nat @ K ) )
= ( ord_less_eq_nat @ K @ I ) ) ).
% atLeast_iff
thf(fact_441_atMost__subset__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X2 ) @ ( set_ord_atMost_set_a @ Y ) )
= ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_442_atMost__subset__iff,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or7210490968680142261et_nat @ X2 ) @ ( set_or7210490968680142261et_nat @ Y ) )
= ( ord_le6893508408891458716et_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_443_atMost__subset__iff,axiom,
! [X2: set_set_b,Y: set_set_b] :
( ( ord_le3201067847557142847_set_b @ ( set_or4087405750901549958_set_b @ X2 ) @ ( set_or4087405750901549958_set_b @ Y ) )
= ( ord_le3795704787696855135_set_b @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_444_atMost__subset__iff,axiom,
! [X2: b,Y: b] :
( ( ord_less_eq_set_b @ ( set_ord_atMost_b @ X2 ) @ ( set_ord_atMost_b @ Y ) )
= ( ord_less_eq_b @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_445_atMost__subset__iff,axiom,
! [X2: a,Y: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ X2 ) @ ( set_ord_atMost_a @ Y ) )
= ( ord_less_eq_a @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_446_atMost__subset__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X2 ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_447_atMost__subset__iff,axiom,
! [X2: set_b,Y: set_b] :
( ( ord_le3795704787696855135_set_b @ ( set_ord_atMost_set_b @ X2 ) @ ( set_ord_atMost_set_b @ Y ) )
= ( ord_less_eq_set_b @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_448_atMost__subset__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_449_atLeast__subset__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or8362275514725411625_set_a @ X2 ) @ ( set_or8362275514725411625_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_450_atLeast__subset__iff,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or1796310902737568945et_nat @ X2 ) @ ( set_or1796310902737568945et_nat @ Y ) )
= ( ord_le6893508408891458716et_nat @ Y @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_451_atLeast__subset__iff,axiom,
! [X2: set_set_b,Y: set_set_b] :
( ( ord_le3201067847557142847_set_b @ ( set_or3975068855832871818_set_b @ X2 ) @ ( set_or3975068855832871818_set_b @ Y ) )
= ( ord_le3795704787696855135_set_b @ Y @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_452_atLeast__subset__iff,axiom,
! [X2: b,Y: b] :
( ( ord_less_eq_set_b @ ( set_ord_atLeast_b @ X2 ) @ ( set_ord_atLeast_b @ Y ) )
= ( ord_less_eq_b @ Y @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_453_atLeast__subset__iff,axiom,
! [X2: a,Y: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ X2 ) @ ( set_ord_atLeast_a @ Y ) )
= ( ord_less_eq_a @ Y @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_454_atLeast__subset__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or1731685050470061051et_nat @ X2 ) @ ( set_or1731685050470061051et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_455_atLeast__subset__iff,axiom,
! [X2: set_b,Y: set_b] :
( ( ord_le3795704787696855135_set_b @ ( set_or8362275519028640426_set_b @ X2 ) @ ( set_or8362275519028640426_set_b @ Y ) )
= ( ord_less_eq_set_b @ Y @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_456_atLeast__subset__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ X2 ) @ ( set_ord_atLeast_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_457_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_458_dense,axiom,
! [X2: a,Y: a] :
( ( ord_less_a @ X2 @ Y )
=> ? [Z3: a] :
( ( ord_less_a @ X2 @ Z3 )
& ( ord_less_a @ Z3 @ Y ) ) ) ).
% dense
thf(fact_459_less__imp__neq,axiom,
! [X2: a,Y: a] :
( ( ord_less_a @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_460_less__imp__neq,axiom,
! [X2: b,Y: b] :
( ( ord_less_b @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_461_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_462_order_Oasym,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order.asym
thf(fact_463_order_Oasym,axiom,
! [A: b,B: b] :
( ( ord_less_b @ A @ B )
=> ~ ( ord_less_b @ B @ A ) ) ).
% order.asym
thf(fact_464_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_465_ord__eq__less__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_466_ord__eq__less__trans,axiom,
! [A: b,B: b,C: b] :
( ( A = B )
=> ( ( ord_less_b @ B @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_467_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_468_ord__less__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_469_ord__less__eq__trans,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_b @ A @ B )
=> ( ( B = C )
=> ( ord_less_b @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_470_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_471_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_472_antisym__conv3,axiom,
! [Y: a,X2: a] :
( ~ ( ord_less_a @ Y @ X2 )
=> ( ( ~ ( ord_less_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_473_antisym__conv3,axiom,
! [Y: b,X2: b] :
( ~ ( ord_less_b @ Y @ X2 )
=> ( ( ~ ( ord_less_b @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_474_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_475_linorder__cases,axiom,
! [X2: a,Y: a] :
( ~ ( ord_less_a @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_a @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_476_linorder__cases,axiom,
! [X2: b,Y: b] :
( ~ ( ord_less_b @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_b @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_477_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_478_dual__order_Oasym,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ~ ( ord_less_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_479_dual__order_Oasym,axiom,
! [B: b,A: b] :
( ( ord_less_b @ B @ A )
=> ~ ( ord_less_b @ A @ B ) ) ).
% dual_order.asym
thf(fact_480_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_481_not__Iic__eq__Ici,axiom,
! [H: nat,L2: nat] :
( ( set_ord_atMost_nat @ H )
!= ( set_ord_atLeast_nat @ L2 ) ) ).
% not_Iic_eq_Ici
thf(fact_482_dual__order_Oirrefl,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_483_dual__order_Oirrefl,axiom,
! [A: b] :
~ ( ord_less_b @ A @ A ) ).
% dual_order.irrefl
thf(fact_484_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_485_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_486_linorder__less__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A4: a,B4: a] :
( ( ord_less_a @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: a] : ( P @ A4 @ A4 )
=> ( ! [A4: a,B4: a] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_487_linorder__less__wlog,axiom,
! [P: b > b > $o,A: b,B: b] :
( ! [A4: b,B4: b] :
( ( ord_less_b @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: b] : ( P @ A4 @ A4 )
=> ( ! [A4: b,B4: b] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_488_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_489_order_Ostrict__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_490_order_Ostrict__trans,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_b @ A @ B )
=> ( ( ord_less_b @ B @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_491_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_492_not__less__iff__gr__or__eq,axiom,
! [X2: a,Y: a] :
( ( ~ ( ord_less_a @ X2 @ Y ) )
= ( ( ord_less_a @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_493_not__less__iff__gr__or__eq,axiom,
! [X2: b,Y: b] :
( ( ~ ( ord_less_b @ X2 @ Y ) )
= ( ( ord_less_b @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_494_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_495_dual__order_Ostrict__trans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_496_dual__order_Ostrict__trans,axiom,
! [B: b,A: b,C: b] :
( ( ord_less_b @ B @ A )
=> ( ( ord_less_b @ C @ B )
=> ( ord_less_b @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_497_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_498_order_Ostrict__implies__not__eq,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_499_order_Ostrict__implies__not__eq,axiom,
! [A: b,B: b] :
( ( ord_less_b @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_500_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_501_dual__order_Ostrict__implies__not__eq,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_502_dual__order_Ostrict__implies__not__eq,axiom,
! [B: b,A: b] :
( ( ord_less_b @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_503_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_504_linorder__neqE,axiom,
! [X2: a,Y: a] :
( ( X2 != Y )
=> ( ~ ( ord_less_a @ X2 @ Y )
=> ( ord_less_a @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_505_linorder__neqE,axiom,
! [X2: b,Y: b] :
( ( X2 != Y )
=> ( ~ ( ord_less_b @ X2 @ Y )
=> ( ord_less_b @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_506_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_507_order__less__asym,axiom,
! [X2: a,Y: a] :
( ( ord_less_a @ X2 @ Y )
=> ~ ( ord_less_a @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_508_order__less__asym,axiom,
! [X2: b,Y: b] :
( ( ord_less_b @ X2 @ Y )
=> ~ ( ord_less_b @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_509_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_510_linorder__neq__iff,axiom,
! [X2: a,Y: a] :
( ( X2 != Y )
= ( ( ord_less_a @ X2 @ Y )
| ( ord_less_a @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_511_linorder__neq__iff,axiom,
! [X2: b,Y: b] :
( ( X2 != Y )
= ( ( ord_less_b @ X2 @ Y )
| ( ord_less_b @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_512_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_513_order__less__asym_H,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ~ ( ord_less_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_514_order__less__asym_H,axiom,
! [A: b,B: b] :
( ( ord_less_b @ A @ B )
=> ~ ( ord_less_b @ B @ A ) ) ).
% order_less_asym'
thf(fact_515_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_516_order__less__trans,axiom,
! [X2: a,Y: a,Z2: a] :
( ( ord_less_a @ X2 @ Y )
=> ( ( ord_less_a @ Y @ Z2 )
=> ( ord_less_a @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_517_order__less__trans,axiom,
! [X2: b,Y: b,Z2: b] :
( ( ord_less_b @ X2 @ Y )
=> ( ( ord_less_b @ Y @ Z2 )
=> ( ord_less_b @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_518_order__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_519_ord__eq__less__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_520_ord__eq__less__subst,axiom,
! [A: b,F: a > b,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_521_ord__eq__less__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_522_ord__eq__less__subst,axiom,
! [A: a,F: b > a,B: b,C: b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_523_ord__eq__less__subst,axiom,
! [A: b,F: b > b,B: b,C: b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_524_ord__eq__less__subst,axiom,
! [A: nat,F: b > nat,B: b,C: b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_525_ord__eq__less__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_526_ord__eq__less__subst,axiom,
! [A: b,F: nat > b,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_527_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_528_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_529_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > b,C: b] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_530_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_531_ord__less__eq__subst,axiom,
! [A: b,B: b,F: b > a,C: a] :
( ( ord_less_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_532_ord__less__eq__subst,axiom,
! [A: b,B: b,F: b > b,C: b] :
( ( ord_less_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_533_ord__less__eq__subst,axiom,
! [A: b,B: b,F: b > nat,C: nat] :
( ( ord_less_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_534_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_535_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > b,C: b] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_536_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_537_order__less__irrefl,axiom,
! [X2: a] :
~ ( ord_less_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_538_order__less__irrefl,axiom,
! [X2: b] :
~ ( ord_less_b @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_539_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_540_order__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_541_order__less__subst1,axiom,
! [A: a,F: b > a,B: b,C: b] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_542_order__less__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_543_order__less__subst1,axiom,
! [A: b,F: a > b,B: a,C: a] :
( ( ord_less_b @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_544_order__less__subst1,axiom,
! [A: b,F: b > b,B: b,C: b] :
( ( ord_less_b @ A @ ( F @ B ) )
=> ( ( ord_less_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_545_order__less__subst1,axiom,
! [A: b,F: nat > b,B: nat,C: nat] :
( ( ord_less_b @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_546_order__less__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_547_order__less__subst1,axiom,
! [A: nat,F: b > nat,B: b,C: b] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_548_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_549_order__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_550_order__less__subst2,axiom,
! [A: a,B: a,F: a > b,C: b] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_b @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_551_order__less__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_552_order__less__subst2,axiom,
! [A: b,B: b,F: b > a,C: a] :
( ( ord_less_b @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_553_order__less__subst2,axiom,
! [A: b,B: b,F: b > b,C: b] :
( ( ord_less_b @ A @ B )
=> ( ( ord_less_b @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_554_order__less__subst2,axiom,
! [A: b,B: b,F: b > nat,C: nat] :
( ( ord_less_b @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_555_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_556_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > b,C: b] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_b @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_557_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_558_order__less__not__sym,axiom,
! [X2: a,Y: a] :
( ( ord_less_a @ X2 @ Y )
=> ~ ( ord_less_a @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_559_order__less__not__sym,axiom,
! [X2: b,Y: b] :
( ( ord_less_b @ X2 @ Y )
=> ~ ( ord_less_b @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_560_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_561_order__less__imp__triv,axiom,
! [X2: a,Y: a,P: $o] :
( ( ord_less_a @ X2 @ Y )
=> ( ( ord_less_a @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_562_order__less__imp__triv,axiom,
! [X2: b,Y: b,P: $o] :
( ( ord_less_b @ X2 @ Y )
=> ( ( ord_less_b @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_563_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_564_linorder__less__linear,axiom,
! [X2: a,Y: a] :
( ( ord_less_a @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_a @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_565_linorder__less__linear,axiom,
! [X2: b,Y: b] :
( ( ord_less_b @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_b @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_566_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_567_order__less__imp__not__eq,axiom,
! [X2: a,Y: a] :
( ( ord_less_a @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_568_order__less__imp__not__eq,axiom,
! [X2: b,Y: b] :
( ( ord_less_b @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_569_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_570_order__less__imp__not__eq2,axiom,
! [X2: a,Y: a] :
( ( ord_less_a @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_571_order__less__imp__not__eq2,axiom,
! [X2: b,Y: b] :
( ( ord_less_b @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_572_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_573_order__less__imp__not__less,axiom,
! [X2: a,Y: a] :
( ( ord_less_a @ X2 @ Y )
=> ~ ( ord_less_a @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_574_order__less__imp__not__less,axiom,
! [X2: b,Y: b] :
( ( ord_less_b @ X2 @ Y )
=> ~ ( ord_less_b @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_575_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_576_not__Ici__le__Iic,axiom,
! [L: nat,H2: nat] :
~ ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ L ) @ ( set_ord_atMost_nat @ H2 ) ) ).
% not_Ici_le_Iic
thf(fact_577_atLeastatMost__psubset__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_less_set_set_a @ ( set_or6288561110385358355_set_a @ A @ B ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_a @ A @ B )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B @ D )
& ( ( ord_less_set_a @ C @ A )
| ( ord_less_set_a @ B @ D ) ) ) )
& ( ord_less_eq_set_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_578_atLeastatMost__psubset__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat,D: set_set_nat] :
( ( ord_le152980574450754630et_nat @ ( set_or9137876137106135879et_nat @ A @ B ) @ ( set_or9137876137106135879et_nat @ C @ D ) )
= ( ( ~ ( ord_le6893508408891458716et_nat @ A @ B )
| ( ( ord_le6893508408891458716et_nat @ C @ A )
& ( ord_le6893508408891458716et_nat @ B @ D )
& ( ( ord_less_set_set_nat @ C @ A )
| ( ord_less_set_set_nat @ B @ D ) ) ) )
& ( ord_le6893508408891458716et_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_579_atLeastatMost__psubset__iff,axiom,
! [A: set_set_b,B: set_set_b,C: set_set_b,D: set_set_b] :
( ( ord_le6262574818192256075_set_b @ ( set_or4832498528752608884_set_b @ A @ B ) @ ( set_or4832498528752608884_set_b @ C @ D ) )
= ( ( ~ ( ord_le3795704787696855135_set_b @ A @ B )
| ( ( ord_le3795704787696855135_set_b @ C @ A )
& ( ord_le3795704787696855135_set_b @ B @ D )
& ( ( ord_less_set_set_b @ C @ A )
| ( ord_less_set_set_b @ B @ D ) ) ) )
& ( ord_le3795704787696855135_set_b @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_580_atLeastatMost__psubset__iff,axiom,
! [A: b,B: b,C: b,D: b] :
( ( ord_less_set_b @ ( set_or672772299803893940Most_b @ A @ B ) @ ( set_or672772299803893940Most_b @ C @ D ) )
= ( ( ~ ( ord_less_eq_b @ A @ B )
| ( ( ord_less_eq_b @ C @ A )
& ( ord_less_eq_b @ B @ D )
& ( ( ord_less_b @ C @ A )
| ( ord_less_b @ B @ D ) ) ) )
& ( ord_less_eq_b @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_581_atLeastatMost__psubset__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_set_a @ ( set_or672772299803893939Most_a @ A @ B ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_a @ A @ B )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D )
& ( ( ord_less_a @ C @ A )
| ( ord_less_a @ B @ D ) ) ) )
& ( ord_less_eq_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_582_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_583_atLeastatMost__psubset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D )
& ( ( ord_less_set_nat @ C @ A )
| ( ord_less_set_nat @ B @ D ) ) ) )
& ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_584_atLeastatMost__psubset__iff,axiom,
! [A: set_b,B: set_b,C: set_b,D: set_b] :
( ( ord_less_set_set_b @ ( set_or6288561114688587156_set_b @ A @ B ) @ ( set_or6288561114688587156_set_b @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_b @ A @ B )
| ( ( ord_less_eq_set_b @ C @ A )
& ( ord_less_eq_set_b @ B @ D )
& ( ( ord_less_set_b @ C @ A )
| ( ord_less_set_b @ B @ D ) ) ) )
& ( ord_less_eq_set_b @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_585_leD,axiom,
! [Y: set_b,X2: set_b] :
( ( ord_less_eq_set_b @ Y @ X2 )
=> ~ ( ord_less_set_b @ X2 @ Y ) ) ).
% leD
thf(fact_586_leD,axiom,
! [Y: a,X2: a] :
( ( ord_less_eq_a @ Y @ X2 )
=> ~ ( ord_less_a @ X2 @ Y ) ) ).
% leD
thf(fact_587_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_588_leD,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y ) ) ).
% leD
thf(fact_589_leD,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ~ ( ord_less_set_a @ X2 @ Y ) ) ).
% leD
thf(fact_590_leD,axiom,
! [Y: b,X2: b] :
( ( ord_less_eq_b @ Y @ X2 )
=> ~ ( ord_less_b @ X2 @ Y ) ) ).
% leD
thf(fact_591_leD,axiom,
! [Y: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ~ ( ord_less_set_set_nat @ X2 @ Y ) ) ).
% leD
thf(fact_592_leD,axiom,
! [Y: set_set_b,X2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ Y @ X2 )
=> ~ ( ord_less_set_set_b @ X2 @ Y ) ) ).
% leD
thf(fact_593_leI,axiom,
! [X2: a,Y: a] :
( ~ ( ord_less_a @ X2 @ Y )
=> ( ord_less_eq_a @ Y @ X2 ) ) ).
% leI
thf(fact_594_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_595_leI,axiom,
! [X2: b,Y: b] :
( ~ ( ord_less_b @ X2 @ Y )
=> ( ord_less_eq_b @ Y @ X2 ) ) ).
% leI
thf(fact_596_nless__le,axiom,
! [A: set_b,B: set_b] :
( ( ~ ( ord_less_set_b @ A @ B ) )
= ( ~ ( ord_less_eq_set_b @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_597_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_598_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_599_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_600_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_601_nless__le,axiom,
! [A: b,B: b] :
( ( ~ ( ord_less_b @ A @ B ) )
= ( ~ ( ord_less_eq_b @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_602_nless__le,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ~ ( ord_less_set_set_nat @ A @ B ) )
= ( ~ ( ord_le6893508408891458716et_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_603_nless__le,axiom,
! [A: set_set_b,B: set_set_b] :
( ( ~ ( ord_less_set_set_b @ A @ B ) )
= ( ~ ( ord_le3795704787696855135_set_b @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_604_antisym__conv1,axiom,
! [X2: set_b,Y: set_b] :
( ~ ( ord_less_set_b @ X2 @ Y )
=> ( ( ord_less_eq_set_b @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_605_antisym__conv1,axiom,
! [X2: a,Y: a] :
( ~ ( ord_less_a @ X2 @ Y )
=> ( ( ord_less_eq_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_606_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_607_antisym__conv1,axiom,
! [X2: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_608_antisym__conv1,axiom,
! [X2: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_609_antisym__conv1,axiom,
! [X2: b,Y: b] :
( ~ ( ord_less_b @ X2 @ Y )
=> ( ( ord_less_eq_b @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_610_antisym__conv1,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ~ ( ord_less_set_set_nat @ X2 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_611_antisym__conv1,axiom,
! [X2: set_set_b,Y: set_set_b] :
( ~ ( ord_less_set_set_b @ X2 @ Y )
=> ( ( ord_le3795704787696855135_set_b @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_612_antisym__conv2,axiom,
! [X2: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y )
=> ( ( ~ ( ord_less_set_b @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_613_antisym__conv2,axiom,
! [X2: a,Y: a] :
( ( ord_less_eq_a @ X2 @ Y )
=> ( ( ~ ( ord_less_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_614_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_615_antisym__conv2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_616_antisym__conv2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ~ ( ord_less_set_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_617_antisym__conv2,axiom,
! [X2: b,Y: b] :
( ( ord_less_eq_b @ X2 @ Y )
=> ( ( ~ ( ord_less_b @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_618_antisym__conv2,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_set_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_619_antisym__conv2,axiom,
! [X2: set_set_b,Y: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X2 @ Y )
=> ( ( ~ ( ord_less_set_set_b @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_620_dense__ge,axiom,
! [Z2: a,Y: a] :
( ! [X: a] :
( ( ord_less_a @ Z2 @ X )
=> ( ord_less_eq_a @ Y @ X ) )
=> ( ord_less_eq_a @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_621_dense__le,axiom,
! [Y: a,Z2: a] :
( ! [X: a] :
( ( ord_less_a @ X @ Y )
=> ( ord_less_eq_a @ X @ Z2 ) )
=> ( ord_less_eq_a @ Y @ Z2 ) ) ).
% dense_le
thf(fact_622_less__le__not__le,axiom,
( ord_less_set_b
= ( ^ [X3: set_b,Y4: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y4 )
& ~ ( ord_less_eq_set_b @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_623_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
& ~ ( ord_less_eq_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_624_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_625_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_626_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_627_less__le__not__le,axiom,
( ord_less_b
= ( ^ [X3: b,Y4: b] :
( ( ord_less_eq_b @ X3 @ Y4 )
& ~ ( ord_less_eq_b @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_628_less__le__not__le,axiom,
( ord_less_set_set_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
& ~ ( ord_le6893508408891458716et_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_629_less__le__not__le,axiom,
( ord_less_set_set_b
= ( ^ [X3: set_set_b,Y4: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y4 )
& ~ ( ord_le3795704787696855135_set_b @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_630_not__le__imp__less,axiom,
! [Y: a,X2: a] :
( ~ ( ord_less_eq_a @ Y @ X2 )
=> ( ord_less_a @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_631_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_632_not__le__imp__less,axiom,
! [Y: b,X2: b] :
( ~ ( ord_less_eq_b @ Y @ X2 )
=> ( ord_less_b @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_633_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_set_b @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_634_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A3: a,B3: a] :
( ( ord_less_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_635_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_636_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_637_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_638_order_Oorder__iff__strict,axiom,
( ord_less_eq_b
= ( ^ [A3: b,B3: b] :
( ( ord_less_b @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_639_order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_640_order_Oorder__iff__strict,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A3: set_set_b,B3: set_set_b] :
( ( ord_less_set_set_b @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_641_order_Ostrict__iff__order,axiom,
( ord_less_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_642_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_643_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_644_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_645_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_646_order_Ostrict__iff__order,axiom,
( ord_less_b
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_647_order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_648_order_Ostrict__iff__order,axiom,
( ord_less_set_set_b
= ( ^ [A3: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_649_order_Ostrict__trans1,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_set_b @ B @ C )
=> ( ord_less_set_b @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_650_order_Ostrict__trans1,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_651_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_652_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_653_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_654_order_Ostrict__trans1,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_b @ B @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_655_order_Ostrict__trans1,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_656_order_Ostrict__trans1,axiom,
! [A: set_set_b,B: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( ord_less_set_set_b @ B @ C )
=> ( ord_less_set_set_b @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_657_order_Ostrict__trans2,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_set_b @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_658_order_Ostrict__trans2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_659_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_660_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_661_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_662_order_Ostrict__trans2,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_b @ A @ B )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_663_order_Ostrict__trans2,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_664_order_Ostrict__trans2,axiom,
! [A: set_set_b,B: set_set_b,C: set_set_b] :
( ( ord_less_set_set_b @ A @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ord_less_set_set_b @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_665_order_Ostrict__iff__not,axiom,
( ord_less_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ~ ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_666_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
& ~ ( ord_less_eq_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_667_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_668_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_669_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_670_order_Ostrict__iff__not,axiom,
( ord_less_b
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ A3 @ B3 )
& ~ ( ord_less_eq_b @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_671_order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ~ ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_672_order_Ostrict__iff__not,axiom,
( ord_less_set_set_b
= ( ^ [A3: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
& ~ ( ord_le3795704787696855135_set_b @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_673_dense__ge__bounded,axiom,
! [Z2: a,X2: a,Y: a] :
( ( ord_less_a @ Z2 @ X2 )
=> ( ! [W: a] :
( ( ord_less_a @ Z2 @ W )
=> ( ( ord_less_a @ W @ X2 )
=> ( ord_less_eq_a @ Y @ W ) ) )
=> ( ord_less_eq_a @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_674_dense__le__bounded,axiom,
! [X2: a,Y: a,Z2: a] :
( ( ord_less_a @ X2 @ Y )
=> ( ! [W: a] :
( ( ord_less_a @ X2 @ W )
=> ( ( ord_less_a @ W @ Y )
=> ( ord_less_eq_a @ W @ Z2 ) ) )
=> ( ord_less_eq_a @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_675_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_b
= ( ^ [B3: set_b,A3: set_b] :
( ( ord_less_set_b @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_676_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B3: a,A3: a] :
( ( ord_less_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_677_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_678_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_679_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_680_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_b
= ( ^ [B3: b,A3: b] :
( ( ord_less_b @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_681_dual__order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B3: set_set_nat,A3: set_set_nat] :
( ( ord_less_set_set_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_682_dual__order_Oorder__iff__strict,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [B3: set_set_b,A3: set_set_b] :
( ( ord_less_set_set_b @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_683_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_b
= ( ^ [B3: set_b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_684_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B3: a,A3: a] :
( ( ord_less_eq_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_685_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_686_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_687_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_688_dual__order_Ostrict__iff__order,axiom,
( ord_less_b
= ( ^ [B3: b,A3: b] :
( ( ord_less_eq_b @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_689_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [B3: set_set_nat,A3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_690_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_b
= ( ^ [B3: set_set_b,A3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_691_dual__order_Ostrict__trans1,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_set_b @ C @ B )
=> ( ord_less_set_b @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_692_dual__order_Ostrict__trans1,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_693_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_694_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_695_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_696_dual__order_Ostrict__trans1,axiom,
! [B: b,A: b,C: b] :
( ( ord_less_eq_b @ B @ A )
=> ( ( ord_less_b @ C @ B )
=> ( ord_less_b @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_697_dual__order_Ostrict__trans1,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_less_set_set_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_698_dual__order_Ostrict__trans1,axiom,
! [B: set_set_b,A: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( ord_less_set_set_b @ C @ B )
=> ( ord_less_set_set_b @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_699_dual__order_Ostrict__trans2,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_set_b @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_700_dual__order_Ostrict__trans2,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_701_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_702_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_703_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_704_dual__order_Ostrict__trans2,axiom,
! [B: b,A: b,C: b] :
( ( ord_less_b @ B @ A )
=> ( ( ord_less_eq_b @ C @ B )
=> ( ord_less_b @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_705_dual__order_Ostrict__trans2,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_706_dual__order_Ostrict__trans2,axiom,
! [B: set_set_b,A: set_set_b,C: set_set_b] :
( ( ord_less_set_set_b @ B @ A )
=> ( ( ord_le3795704787696855135_set_b @ C @ B )
=> ( ord_less_set_set_b @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_707_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_b
= ( ^ [B3: set_b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
& ~ ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_708_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B3: a,A3: a] :
( ( ord_less_eq_a @ B3 @ A3 )
& ~ ( ord_less_eq_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_709_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_710_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ~ ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_711_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ~ ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_712_dual__order_Ostrict__iff__not,axiom,
( ord_less_b
= ( ^ [B3: b,A3: b] :
( ( ord_less_eq_b @ B3 @ A3 )
& ~ ( ord_less_eq_b @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_713_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [B3: set_set_nat,A3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
& ~ ( ord_le6893508408891458716et_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_714_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_b
= ( ^ [B3: set_set_b,A3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B3 @ A3 )
& ~ ( ord_le3795704787696855135_set_b @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_715_order_Ostrict__implies__order,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_set_b @ A @ B )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_716_order_Ostrict__implies__order,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_eq_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_717_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_718_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_719_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_720_order_Ostrict__implies__order,axiom,
! [A: b,B: b] :
( ( ord_less_b @ A @ B )
=> ( ord_less_eq_b @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_721_order_Ostrict__implies__order,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_722_order_Ostrict__implies__order,axiom,
! [A: set_set_b,B: set_set_b] :
( ( ord_less_set_set_b @ A @ B )
=> ( ord_le3795704787696855135_set_b @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_723_dual__order_Ostrict__implies__order,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_set_b @ B @ A )
=> ( ord_less_eq_set_b @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_724_dual__order_Ostrict__implies__order,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_eq_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_725_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_726_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_727_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_728_dual__order_Ostrict__implies__order,axiom,
! [B: b,A: b] :
( ( ord_less_b @ B @ A )
=> ( ord_less_eq_b @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_729_dual__order_Ostrict__implies__order,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_730_dual__order_Ostrict__implies__order,axiom,
! [B: set_set_b,A: set_set_b] :
( ( ord_less_set_set_b @ B @ A )
=> ( ord_le3795704787696855135_set_b @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_731_order__le__less,axiom,
( ord_less_eq_set_b
= ( ^ [X3: set_b,Y4: set_b] :
( ( ord_less_set_b @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_732_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X3: a,Y4: a] :
( ( ord_less_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_733_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_734_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_735_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_set_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_736_order__le__less,axiom,
( ord_less_eq_b
= ( ^ [X3: b,Y4: b] :
( ( ord_less_b @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_737_order__le__less,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_less_set_set_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_738_order__le__less,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [X3: set_set_b,Y4: set_set_b] :
( ( ord_less_set_set_b @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_739_order__less__le,axiom,
( ord_less_set_b
= ( ^ [X3: set_b,Y4: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_740_order__less__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_741_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_742_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_743_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_744_order__less__le,axiom,
( ord_less_b
= ( ^ [X3: b,Y4: b] :
( ( ord_less_eq_b @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_745_order__less__le,axiom,
( ord_less_set_set_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_746_order__less__le,axiom,
( ord_less_set_set_b
= ( ^ [X3: set_set_b,Y4: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_747_linorder__not__le,axiom,
! [X2: a,Y: a] :
( ( ~ ( ord_less_eq_a @ X2 @ Y ) )
= ( ord_less_a @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_748_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_749_linorder__not__le,axiom,
! [X2: b,Y: b] :
( ( ~ ( ord_less_eq_b @ X2 @ Y ) )
= ( ord_less_b @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_750_linorder__not__less,axiom,
! [X2: a,Y: a] :
( ( ~ ( ord_less_a @ X2 @ Y ) )
= ( ord_less_eq_a @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_751_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_752_linorder__not__less,axiom,
! [X2: b,Y: b] :
( ( ~ ( ord_less_b @ X2 @ Y ) )
= ( ord_less_eq_b @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_753_order__less__imp__le,axiom,
! [X2: set_b,Y: set_b] :
( ( ord_less_set_b @ X2 @ Y )
=> ( ord_less_eq_set_b @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_754_order__less__imp__le,axiom,
! [X2: a,Y: a] :
( ( ord_less_a @ X2 @ Y )
=> ( ord_less_eq_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_755_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_756_order__less__imp__le,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_757_order__less__imp__le,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_758_order__less__imp__le,axiom,
! [X2: b,Y: b] :
( ( ord_less_b @ X2 @ Y )
=> ( ord_less_eq_b @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_759_order__less__imp__le,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ Y )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_760_order__less__imp__le,axiom,
! [X2: set_set_b,Y: set_set_b] :
( ( ord_less_set_set_b @ X2 @ Y )
=> ( ord_le3795704787696855135_set_b @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_761_order__le__neq__trans,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_b @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_762_order__le__neq__trans,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_763_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_764_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_765_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_766_order__le__neq__trans,axiom,
! [A: b,B: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( A != B )
=> ( ord_less_b @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_767_order__le__neq__trans,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_768_order__le__neq__trans,axiom,
! [A: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_b @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_769_order__neq__le__trans,axiom,
! [A: set_b,B: set_b] :
( ( A != B )
=> ( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_set_b @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_770_order__neq__le__trans,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_771_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_772_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_773_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_774_order__neq__le__trans,axiom,
! [A: b,B: b] :
( ( A != B )
=> ( ( ord_less_eq_b @ A @ B )
=> ( ord_less_b @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_775_order__neq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A != B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_776_order__neq__le__trans,axiom,
! [A: set_set_b,B: set_set_b] :
( ( A != B )
=> ( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ord_less_set_set_b @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_777_order__le__less__trans,axiom,
! [X2: set_b,Y: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y )
=> ( ( ord_less_set_b @ Y @ Z2 )
=> ( ord_less_set_b @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_778_order__le__less__trans,axiom,
! [X2: a,Y: a,Z2: a] :
( ( ord_less_eq_a @ X2 @ Y )
=> ( ( ord_less_a @ Y @ Z2 )
=> ( ord_less_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_779_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_780_order__le__less__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_781_order__le__less__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_782_order__le__less__trans,axiom,
! [X2: b,Y: b,Z2: b] :
( ( ord_less_eq_b @ X2 @ Y )
=> ( ( ord_less_b @ Y @ Z2 )
=> ( ord_less_b @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_783_order__le__less__trans,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ord_less_set_set_nat @ Y @ Z2 )
=> ( ord_less_set_set_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_784_order__le__less__trans,axiom,
! [X2: set_set_b,Y: set_set_b,Z2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X2 @ Y )
=> ( ( ord_less_set_set_b @ Y @ Z2 )
=> ( ord_less_set_set_b @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_785_order__less__le__trans,axiom,
! [X2: set_b,Y: set_b,Z2: set_b] :
( ( ord_less_set_b @ X2 @ Y )
=> ( ( ord_less_eq_set_b @ Y @ Z2 )
=> ( ord_less_set_b @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_786_order__less__le__trans,axiom,
! [X2: a,Y: a,Z2: a] :
( ( ord_less_a @ X2 @ Y )
=> ( ( ord_less_eq_a @ Y @ Z2 )
=> ( ord_less_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_787_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_788_order__less__le__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_789_order__less__le__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_790_order__less__le__trans,axiom,
! [X2: b,Y: b,Z2: b] :
( ( ord_less_b @ X2 @ Y )
=> ( ( ord_less_eq_b @ Y @ Z2 )
=> ( ord_less_b @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_791_order__less__le__trans,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z2 )
=> ( ord_less_set_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_792_order__less__le__trans,axiom,
! [X2: set_set_b,Y: set_set_b,Z2: set_set_b] :
( ( ord_less_set_set_b @ X2 @ Y )
=> ( ( ord_le3795704787696855135_set_b @ Y @ Z2 )
=> ( ord_less_set_set_b @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_793_order__le__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_794_order__le__less__subst1,axiom,
! [A: a,F: b > a,B: b,C: b] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_795_order__le__less__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_796_order__le__less__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_797_order__le__less__subst1,axiom,
! [A: nat,F: b > nat,B: b,C: b] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_798_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_799_order__le__less__subst1,axiom,
! [A: b,F: a > b,B: a,C: a] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_800_order__le__less__subst1,axiom,
! [A: b,F: b > b,B: b,C: b] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_801_order__le__less__subst1,axiom,
! [A: b,F: nat > b,B: nat,C: nat] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_802_order__le__less__subst1,axiom,
! [A: set_b,F: a > set_b,B: a,C: a] :
( ( ord_less_eq_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_803_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_804_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_805_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > b,C: b] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_b @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_806_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_807_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_808_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_b @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_809_order__le__less__subst2,axiom,
! [A: b,B: b,F: b > a,C: a] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_810_order__le__less__subst2,axiom,
! [A: b,B: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_811_order__le__less__subst2,axiom,
! [A: b,B: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_b @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_812_order__le__less__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > a,C: a] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_813_order__less__le__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_814_order__less__le__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_815_order__less__le__subst1,axiom,
! [A: b,F: a > b,B: a,C: a] :
( ( ord_less_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_816_order__less__le__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_817_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_818_order__less__le__subst1,axiom,
! [A: b,F: nat > b,B: nat,C: nat] :
( ( ord_less_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_819_order__less__le__subst1,axiom,
! [A: a,F: b > a,B: b,C: b] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_820_order__less__le__subst1,axiom,
! [A: nat,F: b > nat,B: b,C: b] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_821_order__less__le__subst1,axiom,
! [A: b,F: b > b,B: b,C: b] :
( ( ord_less_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_eq_b @ X @ Y2 )
=> ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_822_order__less__le__subst1,axiom,
! [A: a,F: set_b > a,B: set_b,C: set_b] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_823_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_824_order__less__le__subst2,axiom,
! [A: b,B: b,F: b > a,C: a] :
( ( ord_less_b @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_825_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_826_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_827_order__less__le__subst2,axiom,
! [A: b,B: b,F: b > nat,C: nat] :
( ( ord_less_b @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_828_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_829_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > b,C: b] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_830_order__less__le__subst2,axiom,
! [A: b,B: b,F: b > b,C: b] :
( ( ord_less_b @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X: b,Y2: b] :
( ( ord_less_b @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_831_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > b,C: b] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_832_order__less__le__subst2,axiom,
! [A: a,B: a,F: a > set_b,C: set_b] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_b @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_833_linorder__le__less__linear,axiom,
! [X2: a,Y: a] :
( ( ord_less_eq_a @ X2 @ Y )
| ( ord_less_a @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_834_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_835_linorder__le__less__linear,axiom,
! [X2: b,Y: b] :
( ( ord_less_eq_b @ X2 @ Y )
| ( ord_less_b @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_836_order__le__imp__less__or__eq,axiom,
! [X2: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y )
=> ( ( ord_less_set_b @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_837_order__le__imp__less__or__eq,axiom,
! [X2: a,Y: a] :
( ( ord_less_eq_a @ X2 @ Y )
=> ( ( ord_less_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_838_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_839_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_840_order__le__imp__less__or__eq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_841_order__le__imp__less__or__eq,axiom,
! [X2: b,Y: b] :
( ( ord_less_eq_b @ X2 @ Y )
=> ( ( ord_less_b @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_842_order__le__imp__less__or__eq,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ord_less_set_set_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_843_order__le__imp__less__or__eq,axiom,
! [X2: set_set_b,Y: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X2 @ Y )
=> ( ( ord_less_set_set_b @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_844_not__Ici__eq__Icc,axiom,
! [L2: nat,L: nat,H: nat] :
( ( set_ord_atLeast_nat @ L2 )
!= ( set_or1269000886237332187st_nat @ L @ H ) ) ).
% not_Ici_eq_Icc
thf(fact_845_not__Ici__le__Icc,axiom,
! [L: nat,L2: nat,H2: nat] :
~ ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ L ) @ ( set_or1269000886237332187st_nat @ L2 @ H2 ) ) ).
% not_Ici_le_Icc
thf(fact_846_ord_Omono__on__subset,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > a,B2: set_b] :
( ( monotone_on_b_a @ A2 @ Less_eq @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( monotone_on_b_a @ B2 @ Less_eq @ ord_less_eq_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_847_ord_Omono__on__subset,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > a,B2: set_nat] :
( ( monotone_on_nat_a @ A2 @ Less_eq @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( monotone_on_nat_a @ B2 @ Less_eq @ ord_less_eq_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_848_ord_Omono__on__subset,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > a,B2: set_a] :
( ( monotone_on_a_a @ A2 @ Less_eq @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( monotone_on_a_a @ B2 @ Less_eq @ ord_less_eq_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_849_ord_Omono__on__subset,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > nat,B2: set_b] :
( ( monotone_on_b_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( monotone_on_b_nat @ B2 @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_850_ord_Omono__on__subset,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat,B2: set_nat] :
( ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( monotone_on_nat_nat @ B2 @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_851_ord_Omono__on__subset,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > nat,B2: set_a] :
( ( monotone_on_a_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( monotone_on_a_nat @ B2 @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_852_ord_Omono__on__subset,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > b,B2: set_b] :
( ( monotone_on_b_b @ A2 @ Less_eq @ ord_less_eq_b @ F )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( monotone_on_b_b @ B2 @ Less_eq @ ord_less_eq_b @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_853_ord_Omono__on__subset,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > b,B2: set_nat] :
( ( monotone_on_nat_b @ A2 @ Less_eq @ ord_less_eq_b @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( monotone_on_nat_b @ B2 @ Less_eq @ ord_less_eq_b @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_854_ord_Omono__on__subset,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > b,B2: set_a] :
( ( monotone_on_a_b @ A2 @ Less_eq @ ord_less_eq_b @ F )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( monotone_on_a_b @ B2 @ Less_eq @ ord_less_eq_b @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_855_ord_Omono__on__subset,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > set_b,B2: set_b] :
( ( monotone_on_b_set_b @ A2 @ Less_eq @ ord_less_eq_set_b @ F )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( monotone_on_b_set_b @ B2 @ Less_eq @ ord_less_eq_set_b @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_856_mono__on__subset,axiom,
! [A2: set_a,F: a > a,B2: set_a] :
( ( monotone_on_a_a @ A2 @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( monotone_on_a_a @ B2 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ) ).
% mono_on_subset
thf(fact_857_mono__on__subset,axiom,
! [A2: set_a,F: a > nat,B2: set_a] :
( ( monotone_on_a_nat @ A2 @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( monotone_on_a_nat @ B2 @ ord_less_eq_a @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_858_mono__on__subset,axiom,
! [A2: set_a,F: a > b,B2: set_a] :
( ( monotone_on_a_b @ A2 @ ord_less_eq_a @ ord_less_eq_b @ F )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( monotone_on_a_b @ B2 @ ord_less_eq_a @ ord_less_eq_b @ F ) ) ) ).
% mono_on_subset
thf(fact_859_mono__on__subset,axiom,
! [A2: set_nat,F: nat > a,B2: set_nat] :
( ( monotone_on_nat_a @ A2 @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( monotone_on_nat_a @ B2 @ ord_less_eq_nat @ ord_less_eq_a @ F ) ) ) ).
% mono_on_subset
thf(fact_860_mono__on__subset,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( monotone_on_nat_nat @ B2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_861_mono__on__subset,axiom,
! [A2: set_nat,F: nat > b,B2: set_nat] :
( ( monotone_on_nat_b @ A2 @ ord_less_eq_nat @ ord_less_eq_b @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( monotone_on_nat_b @ B2 @ ord_less_eq_nat @ ord_less_eq_b @ F ) ) ) ).
% mono_on_subset
thf(fact_862_mono__on__subset,axiom,
! [A2: set_b,F: b > a,B2: set_b] :
( ( monotone_on_b_a @ A2 @ ord_less_eq_b @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( monotone_on_b_a @ B2 @ ord_less_eq_b @ ord_less_eq_a @ F ) ) ) ).
% mono_on_subset
thf(fact_863_mono__on__subset,axiom,
! [A2: set_b,F: b > nat,B2: set_b] :
( ( monotone_on_b_nat @ A2 @ ord_less_eq_b @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( monotone_on_b_nat @ B2 @ ord_less_eq_b @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_864_mono__on__subset,axiom,
! [A2: set_b,F: b > b,B2: set_b] :
( ( monotone_on_b_b @ A2 @ ord_less_eq_b @ ord_less_eq_b @ F )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( monotone_on_b_b @ B2 @ ord_less_eq_b @ ord_less_eq_b @ F ) ) ) ).
% mono_on_subset
thf(fact_865_mono__on__subset,axiom,
! [A2: set_set_b,F: set_b > a,B2: set_set_b] :
( ( monotone_on_set_b_a @ A2 @ ord_less_eq_set_b @ ord_less_eq_a @ F )
=> ( ( ord_le3795704787696855135_set_b @ B2 @ A2 )
=> ( monotone_on_set_b_a @ B2 @ ord_less_eq_set_b @ ord_less_eq_a @ F ) ) ) ).
% mono_on_subset
thf(fact_866_mono__on__greaterD,axiom,
! [A2: set_a,G: a > a,X2: a,Y: a] :
( ( monotone_on_a_a @ A2 @ ord_less_eq_a @ ord_less_eq_a @ G )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ord_less_a @ ( G @ Y ) @ ( G @ X2 ) )
=> ( ord_less_a @ Y @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_867_mono__on__greaterD,axiom,
! [A2: set_a,G: a > nat,X2: a,Y: a] :
( ( monotone_on_a_nat @ A2 @ ord_less_eq_a @ ord_less_eq_nat @ G )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X2 ) )
=> ( ord_less_a @ Y @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_868_mono__on__greaterD,axiom,
! [A2: set_a,G: a > b,X2: a,Y: a] :
( ( monotone_on_a_b @ A2 @ ord_less_eq_a @ ord_less_eq_b @ G )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ord_less_b @ ( G @ Y ) @ ( G @ X2 ) )
=> ( ord_less_a @ Y @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_869_mono__on__greaterD,axiom,
! [A2: set_nat,G: nat > a,X2: nat,Y: nat] :
( ( monotone_on_nat_a @ A2 @ ord_less_eq_nat @ ord_less_eq_a @ G )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_a @ ( G @ Y ) @ ( G @ X2 ) )
=> ( ord_less_nat @ Y @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_870_mono__on__greaterD,axiom,
! [A2: set_nat,G: nat > nat,X2: nat,Y: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ G )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X2 ) )
=> ( ord_less_nat @ Y @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_871_mono__on__greaterD,axiom,
! [A2: set_nat,G: nat > b,X2: nat,Y: nat] :
( ( monotone_on_nat_b @ A2 @ ord_less_eq_nat @ ord_less_eq_b @ G )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_b @ ( G @ Y ) @ ( G @ X2 ) )
=> ( ord_less_nat @ Y @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_872_mono__on__greaterD,axiom,
! [A2: set_b,G: b > a,X2: b,Y: b] :
( ( monotone_on_b_a @ A2 @ ord_less_eq_b @ ord_less_eq_a @ G )
=> ( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( ( ord_less_a @ ( G @ Y ) @ ( G @ X2 ) )
=> ( ord_less_b @ Y @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_873_mono__on__greaterD,axiom,
! [A2: set_b,G: b > nat,X2: b,Y: b] :
( ( monotone_on_b_nat @ A2 @ ord_less_eq_b @ ord_less_eq_nat @ G )
=> ( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X2 ) )
=> ( ord_less_b @ Y @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_874_mono__on__greaterD,axiom,
! [A2: set_b,G: b > b,X2: b,Y: b] :
( ( monotone_on_b_b @ A2 @ ord_less_eq_b @ ord_less_eq_b @ G )
=> ( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( ( ord_less_b @ ( G @ Y ) @ ( G @ X2 ) )
=> ( ord_less_b @ Y @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_875_strict__mono__on__leD,axiom,
! [A2: set_a,F: a > a,X2: a,Y: a] :
( ( monotone_on_a_a @ A2 @ ord_less_a @ ord_less_a @ F )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ord_less_eq_a @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_876_strict__mono__on__leD,axiom,
! [A2: set_a,F: a > nat,X2: a,Y: a] :
( ( monotone_on_a_nat @ A2 @ ord_less_a @ ord_less_nat @ F )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ord_less_eq_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_877_strict__mono__on__leD,axiom,
! [A2: set_a,F: a > b,X2: a,Y: a] :
( ( monotone_on_a_b @ A2 @ ord_less_a @ ord_less_b @ F )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ord_less_eq_a @ X2 @ Y )
=> ( ord_less_eq_b @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_878_strict__mono__on__leD,axiom,
! [A2: set_nat,F: nat > a,X2: nat,Y: nat] :
( ( monotone_on_nat_a @ A2 @ ord_less_nat @ ord_less_a @ F )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_879_strict__mono__on__leD,axiom,
! [A2: set_nat,F: nat > nat,X2: nat,Y: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_880_strict__mono__on__leD,axiom,
! [A2: set_nat,F: nat > b,X2: nat,Y: nat] :
( ( monotone_on_nat_b @ A2 @ ord_less_nat @ ord_less_b @ F )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_b @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_881_strict__mono__on__leD,axiom,
! [A2: set_b,F: b > a,X2: b,Y: b] :
( ( monotone_on_b_a @ A2 @ ord_less_b @ ord_less_a @ F )
=> ( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( ( ord_less_eq_b @ X2 @ Y )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_882_strict__mono__on__leD,axiom,
! [A2: set_b,F: b > nat,X2: b,Y: b] :
( ( monotone_on_b_nat @ A2 @ ord_less_b @ ord_less_nat @ F )
=> ( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( ( ord_less_eq_b @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_883_strict__mono__on__leD,axiom,
! [A2: set_b,F: b > b,X2: b,Y: b] :
( ( monotone_on_b_b @ A2 @ ord_less_b @ ord_less_b @ F )
=> ( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( ( ord_less_eq_b @ X2 @ Y )
=> ( ord_less_eq_b @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_884_strict__mono__on__leD,axiom,
! [A2: set_a,F: a > set_b,X2: a,Y: a] :
( ( monotone_on_a_set_b @ A2 @ ord_less_a @ ord_less_set_b @ F )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ord_less_eq_a @ X2 @ Y )
=> ( ord_less_eq_set_b @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_885_strict__mono__on__imp__mono__on,axiom,
! [A2: set_a,F: a > a] :
( ( monotone_on_a_a @ A2 @ ord_less_a @ ord_less_a @ F )
=> ( monotone_on_a_a @ A2 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_886_strict__mono__on__imp__mono__on,axiom,
! [A2: set_a,F: a > nat] :
( ( monotone_on_a_nat @ A2 @ ord_less_a @ ord_less_nat @ F )
=> ( monotone_on_a_nat @ A2 @ ord_less_eq_a @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_887_strict__mono__on__imp__mono__on,axiom,
! [A2: set_a,F: a > b] :
( ( monotone_on_a_b @ A2 @ ord_less_a @ ord_less_b @ F )
=> ( monotone_on_a_b @ A2 @ ord_less_eq_a @ ord_less_eq_b @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_888_strict__mono__on__imp__mono__on,axiom,
! [A2: set_nat,F: nat > a] :
( ( monotone_on_nat_a @ A2 @ ord_less_nat @ ord_less_a @ F )
=> ( monotone_on_nat_a @ A2 @ ord_less_eq_nat @ ord_less_eq_a @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_889_strict__mono__on__imp__mono__on,axiom,
! [A2: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_890_strict__mono__on__imp__mono__on,axiom,
! [A2: set_nat,F: nat > b] :
( ( monotone_on_nat_b @ A2 @ ord_less_nat @ ord_less_b @ F )
=> ( monotone_on_nat_b @ A2 @ ord_less_eq_nat @ ord_less_eq_b @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_891_strict__mono__on__imp__mono__on,axiom,
! [A2: set_b,F: b > a] :
( ( monotone_on_b_a @ A2 @ ord_less_b @ ord_less_a @ F )
=> ( monotone_on_b_a @ A2 @ ord_less_eq_b @ ord_less_eq_a @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_892_strict__mono__on__imp__mono__on,axiom,
! [A2: set_b,F: b > nat] :
( ( monotone_on_b_nat @ A2 @ ord_less_b @ ord_less_nat @ F )
=> ( monotone_on_b_nat @ A2 @ ord_less_eq_b @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_893_strict__mono__on__imp__mono__on,axiom,
! [A2: set_b,F: b > b] :
( ( monotone_on_b_b @ A2 @ ord_less_b @ ord_less_b @ F )
=> ( monotone_on_b_b @ A2 @ ord_less_eq_b @ ord_less_eq_b @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_894_strict__mono__on__imp__mono__on,axiom,
! [A2: set_a,F: a > set_b] :
( ( monotone_on_a_set_b @ A2 @ ord_less_a @ ord_less_set_b @ F )
=> ( monotone_on_a_set_b @ A2 @ ord_less_eq_a @ ord_less_eq_set_b @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_895_monotone__on__subset,axiom,
! [A2: set_a,Orda: a > a > $o,Ordb: b > b > $o,F: a > b,B2: set_a] :
( ( monotone_on_a_b @ A2 @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( monotone_on_a_b @ B2 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_896_strict__mono__on__eqD,axiom,
! [A2: set_a,F: a > a,X2: a,Y: a] :
( ( monotone_on_a_a @ A2 @ ord_less_a @ ord_less_a @ F )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( Y = X2 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_897_strict__mono__on__eqD,axiom,
! [A2: set_a,F: a > b,X2: a,Y: a] :
( ( monotone_on_a_b @ A2 @ ord_less_a @ ord_less_b @ F )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( Y = X2 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_898_strict__mono__on__eqD,axiom,
! [A2: set_a,F: a > nat,X2: a,Y: a] :
( ( monotone_on_a_nat @ A2 @ ord_less_a @ ord_less_nat @ F )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( Y = X2 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_899_strict__mono__on__eqD,axiom,
! [A2: set_b,F: b > a,X2: b,Y: b] :
( ( monotone_on_b_a @ A2 @ ord_less_b @ ord_less_a @ F )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( Y = X2 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_900_strict__mono__on__eqD,axiom,
! [A2: set_b,F: b > b,X2: b,Y: b] :
( ( monotone_on_b_b @ A2 @ ord_less_b @ ord_less_b @ F )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( Y = X2 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_901_strict__mono__on__eqD,axiom,
! [A2: set_b,F: b > nat,X2: b,Y: b] :
( ( monotone_on_b_nat @ A2 @ ord_less_b @ ord_less_nat @ F )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( Y = X2 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_902_strict__mono__on__eqD,axiom,
! [A2: set_nat,F: nat > a,X2: nat,Y: nat] :
( ( monotone_on_nat_a @ A2 @ ord_less_nat @ ord_less_a @ F )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( Y = X2 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_903_strict__mono__on__eqD,axiom,
! [A2: set_nat,F: nat > b,X2: nat,Y: nat] :
( ( monotone_on_nat_b @ A2 @ ord_less_nat @ ord_less_b @ F )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( Y = X2 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_904_strict__mono__on__eqD,axiom,
! [A2: set_nat,F: nat > nat,X2: nat,Y: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( Y = X2 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_905_strict__mono__onI,axiom,
! [A2: set_a,F: a > a] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( ord_less_a @ R2 @ S3 )
=> ( ord_less_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_a @ A2 @ ord_less_a @ ord_less_a @ F ) ) ).
% strict_mono_onI
thf(fact_906_strict__mono__onI,axiom,
! [A2: set_a,F: a > b] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( ord_less_a @ R2 @ S3 )
=> ( ord_less_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_b @ A2 @ ord_less_a @ ord_less_b @ F ) ) ).
% strict_mono_onI
thf(fact_907_strict__mono__onI,axiom,
! [A2: set_a,F: a > nat] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( ord_less_a @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_nat @ A2 @ ord_less_a @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_908_strict__mono__onI,axiom,
! [A2: set_b,F: b > a] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( ord_less_b @ R2 @ S3 )
=> ( ord_less_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_a @ A2 @ ord_less_b @ ord_less_a @ F ) ) ).
% strict_mono_onI
thf(fact_909_strict__mono__onI,axiom,
! [A2: set_b,F: b > b] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( ord_less_b @ R2 @ S3 )
=> ( ord_less_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_b @ A2 @ ord_less_b @ ord_less_b @ F ) ) ).
% strict_mono_onI
thf(fact_910_strict__mono__onI,axiom,
! [A2: set_b,F: b > nat] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( ord_less_b @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_nat @ A2 @ ord_less_b @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_911_strict__mono__onI,axiom,
! [A2: set_nat,F: nat > a] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_nat @ R2 @ S3 )
=> ( ord_less_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_a @ A2 @ ord_less_nat @ ord_less_a @ F ) ) ).
% strict_mono_onI
thf(fact_912_strict__mono__onI,axiom,
! [A2: set_nat,F: nat > b] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_nat @ R2 @ S3 )
=> ( ord_less_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_b @ A2 @ ord_less_nat @ ord_less_b @ F ) ) ).
% strict_mono_onI
thf(fact_913_strict__mono__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_nat @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_914_strict__mono__onI,axiom,
! [A2: set_set_nat,F: set_nat > a] :
( ! [R2: set_nat,S3: set_nat] :
( ( member_set_nat @ R2 @ A2 )
=> ( ( member_set_nat @ S3 @ A2 )
=> ( ( ord_less_set_nat @ R2 @ S3 )
=> ( ord_less_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto2395835772568396751_nat_a @ A2 @ ord_less_set_nat @ ord_less_a @ F ) ) ).
% strict_mono_onI
thf(fact_915_strict__mono__onD,axiom,
! [A2: set_a,F: a > a,R: a,S2: a] :
( ( monotone_on_a_a @ A2 @ ord_less_a @ ord_less_a @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( ord_less_a @ R @ S2 )
=> ( ord_less_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_916_strict__mono__onD,axiom,
! [A2: set_a,F: a > b,R: a,S2: a] :
( ( monotone_on_a_b @ A2 @ ord_less_a @ ord_less_b @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( ord_less_a @ R @ S2 )
=> ( ord_less_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_917_strict__mono__onD,axiom,
! [A2: set_a,F: a > nat,R: a,S2: a] :
( ( monotone_on_a_nat @ A2 @ ord_less_a @ ord_less_nat @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( ord_less_a @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_918_strict__mono__onD,axiom,
! [A2: set_b,F: b > a,R: b,S2: b] :
( ( monotone_on_b_a @ A2 @ ord_less_b @ ord_less_a @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( ord_less_b @ R @ S2 )
=> ( ord_less_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_919_strict__mono__onD,axiom,
! [A2: set_b,F: b > b,R: b,S2: b] :
( ( monotone_on_b_b @ A2 @ ord_less_b @ ord_less_b @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( ord_less_b @ R @ S2 )
=> ( ord_less_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_920_strict__mono__onD,axiom,
! [A2: set_b,F: b > nat,R: b,S2: b] :
( ( monotone_on_b_nat @ A2 @ ord_less_b @ ord_less_nat @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( ord_less_b @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_921_strict__mono__onD,axiom,
! [A2: set_nat,F: nat > a,R: nat,S2: nat] :
( ( monotone_on_nat_a @ A2 @ ord_less_nat @ ord_less_a @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_nat @ R @ S2 )
=> ( ord_less_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_922_strict__mono__onD,axiom,
! [A2: set_nat,F: nat > b,R: nat,S2: nat] :
( ( monotone_on_nat_b @ A2 @ ord_less_nat @ ord_less_b @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_nat @ R @ S2 )
=> ( ord_less_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_923_strict__mono__onD,axiom,
! [A2: set_nat,F: nat > nat,R: nat,S2: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_nat @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_924_strict__mono__onD,axiom,
! [A2: set_set_nat,F: set_nat > a,R: set_nat,S2: set_nat] :
( ( monoto2395835772568396751_nat_a @ A2 @ ord_less_set_nat @ ord_less_a @ F )
=> ( ( member_set_nat @ R @ A2 )
=> ( ( member_set_nat @ S2 @ A2 )
=> ( ( ord_less_set_nat @ R @ S2 )
=> ( ord_less_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_925_ord_Ostrict__mono__on__def,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > a] :
( ( monotone_on_nat_a @ A2 @ Less @ ord_less_a @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_a @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_926_ord_Ostrict__mono__on__def,axiom,
! [A2: set_a,Less: a > a > $o,F: a > a] :
( ( monotone_on_a_a @ A2 @ Less @ ord_less_a @ F )
= ( ! [R3: a,S4: a] :
( ( ( member_a @ R3 @ A2 )
& ( member_a @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_a @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_927_ord_Ostrict__mono__on__def,axiom,
! [A2: set_b,Less: b > b > $o,F: b > a] :
( ( monotone_on_b_a @ A2 @ Less @ ord_less_a @ F )
= ( ! [R3: b,S4: b] :
( ( ( member_b @ R3 @ A2 )
& ( member_b @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_a @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_928_ord_Ostrict__mono__on__def,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > b] :
( ( monotone_on_nat_b @ A2 @ Less @ ord_less_b @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_b @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_929_ord_Ostrict__mono__on__def,axiom,
! [A2: set_b,Less: b > b > $o,F: b > b] :
( ( monotone_on_b_b @ A2 @ Less @ ord_less_b @ F )
= ( ! [R3: b,S4: b] :
( ( ( member_b @ R3 @ A2 )
& ( member_b @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_b @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_930_ord_Ostrict__mono__on__def,axiom,
! [A2: set_a,Less: a > a > $o,F: a > b] :
( ( monotone_on_a_b @ A2 @ Less @ ord_less_b @ F )
= ( ! [R3: a,S4: a] :
( ( ( member_a @ R3 @ A2 )
& ( member_a @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_b @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_931_ord_Ostrict__mono__on__def,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A2 @ Less @ ord_less_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_932_ord_Ostrict__mono__on__def,axiom,
! [A2: set_a,Less: a > a > $o,F: a > nat] :
( ( monotone_on_a_nat @ A2 @ Less @ ord_less_nat @ F )
= ( ! [R3: a,S4: a] :
( ( ( member_a @ R3 @ A2 )
& ( member_a @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_933_ord_Ostrict__mono__on__def,axiom,
! [A2: set_b,Less: b > b > $o,F: b > nat] :
( ( monotone_on_b_nat @ A2 @ Less @ ord_less_nat @ F )
= ( ! [R3: b,S4: b] :
( ( ( member_b @ R3 @ A2 )
& ( member_b @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_934_ord_Ostrict__mono__on__def,axiom,
! [A2: set_set_nat,Less: set_nat > set_nat > $o,F: set_nat > a] :
( ( monoto2395835772568396751_nat_a @ A2 @ Less @ ord_less_a @ F )
= ( ! [R3: set_nat,S4: set_nat] :
( ( ( member_set_nat @ R3 @ A2 )
& ( member_set_nat @ S4 @ A2 )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_a @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_935_psubsetI,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_b @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_936_psubsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_937_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_938_psubsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_939_psubsetI,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_b @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_940_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_941_psubsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_942_psubsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_943_psubsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_944_psubsetD,axiom,
! [A2: set_set_b,B2: set_set_b,C: set_b] :
( ( ord_less_set_set_b @ A2 @ B2 )
=> ( ( member_set_b @ C @ A2 )
=> ( member_set_b @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_945_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B5: set_b] :
( ( ord_less_set_b @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_946_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_947_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_948_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_less_set_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_949_subset__iff__psubset__eq,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A5: set_set_b,B5: set_set_b] :
( ( ord_less_set_set_b @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_950_subset__psubset__trans,axiom,
! [A2: set_b,B2: set_b,C3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_set_b @ B2 @ C3 )
=> ( ord_less_set_b @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_951_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C3 )
=> ( ord_less_set_nat @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_952_subset__psubset__trans,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C3 )
=> ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_953_subset__psubset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_set_set_nat @ B2 @ C3 )
=> ( ord_less_set_set_nat @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_954_subset__psubset__trans,axiom,
! [A2: set_set_b,B2: set_set_b,C3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ( ( ord_less_set_set_b @ B2 @ C3 )
=> ( ord_less_set_set_b @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_955_subset__not__subset__eq,axiom,
( ord_less_set_b
= ( ^ [A5: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ A5 @ B5 )
& ~ ( ord_less_eq_set_b @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_956_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_957_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_958_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B5 )
& ~ ( ord_le6893508408891458716et_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_959_subset__not__subset__eq,axiom,
( ord_less_set_set_b
= ( ^ [A5: set_set_b,B5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A5 @ B5 )
& ~ ( ord_le3795704787696855135_set_b @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_960_psubset__subset__trans,axiom,
! [A2: set_b,B2: set_b,C3: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C3 )
=> ( ord_less_set_b @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_961_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ord_less_set_nat @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_962_psubset__subset__trans,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_963_psubset__subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C3: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C3 )
=> ( ord_less_set_set_nat @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_964_psubset__subset__trans,axiom,
! [A2: set_set_b,B2: set_set_b,C3: set_set_b] :
( ( ord_less_set_set_b @ A2 @ B2 )
=> ( ( ord_le3795704787696855135_set_b @ B2 @ C3 )
=> ( ord_less_set_set_b @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_965_psubset__imp__subset,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_966_psubset__imp__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_967_psubset__imp__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_968_psubset__imp__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_969_psubset__imp__subset,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( ord_less_set_set_b @ A2 @ B2 )
=> ( ord_le3795704787696855135_set_b @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_970_psubset__eq,axiom,
( ord_less_set_b
= ( ^ [A5: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_971_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_972_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_973_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_974_psubset__eq,axiom,
( ord_less_set_set_b
= ( ^ [A5: set_set_b,B5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_975_psubsetE,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_976_psubsetE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_977_psubsetE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_978_psubsetE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_979_psubsetE,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( ord_less_set_set_b @ A2 @ B2 )
=> ~ ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ( ord_le3795704787696855135_set_b @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_980_monotone__onD,axiom,
! [A2: set_a,Orda: a > a > $o,Ordb: b > b > $o,F: a > b,X2: a,Y: a] :
( ( monotone_on_a_b @ A2 @ Orda @ Ordb @ F )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( Orda @ X2 @ Y )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_981_monotone__onI,axiom,
! [A2: set_a,Orda: a > a > $o,Ordb: b > b > $o,F: a > b] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ A2 )
=> ( ( member_a @ Y2 @ A2 )
=> ( ( Orda @ X @ Y2 )
=> ( Ordb @ ( F @ X ) @ ( F @ Y2 ) ) ) ) )
=> ( monotone_on_a_b @ A2 @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_982_monotone__on__def,axiom,
( monotone_on_a_b
= ( ^ [A5: set_a,Orda2: a > a > $o,Ordb2: b > b > $o,F3: a > b] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ A5 )
=> ( ( Orda2 @ X3 @ Y4 )
=> ( Ordb2 @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_983_mono__onI,axiom,
! [A2: set_a,F: a > a] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( ord_less_eq_a @ R2 @ S3 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_a @ A2 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% mono_onI
thf(fact_984_mono__onI,axiom,
! [A2: set_a,F: a > nat] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( ord_less_eq_a @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_nat @ A2 @ ord_less_eq_a @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_985_mono__onI,axiom,
! [A2: set_a,F: a > b] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( ord_less_eq_a @ R2 @ S3 )
=> ( ord_less_eq_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_b @ A2 @ ord_less_eq_a @ ord_less_eq_b @ F ) ) ).
% mono_onI
thf(fact_986_mono__onI,axiom,
! [A2: set_nat,F: nat > a] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_a @ A2 @ ord_less_eq_nat @ ord_less_eq_a @ F ) ) ).
% mono_onI
thf(fact_987_mono__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_988_mono__onI,axiom,
! [A2: set_nat,F: nat > b] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_less_eq_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_b @ A2 @ ord_less_eq_nat @ ord_less_eq_b @ F ) ) ).
% mono_onI
thf(fact_989_mono__onI,axiom,
! [A2: set_b,F: b > a] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( ord_less_eq_b @ R2 @ S3 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_a @ A2 @ ord_less_eq_b @ ord_less_eq_a @ F ) ) ).
% mono_onI
thf(fact_990_mono__onI,axiom,
! [A2: set_b,F: b > nat] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( ord_less_eq_b @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_nat @ A2 @ ord_less_eq_b @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_991_mono__onI,axiom,
! [A2: set_b,F: b > b] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( ord_less_eq_b @ R2 @ S3 )
=> ( ord_less_eq_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_b @ A2 @ ord_less_eq_b @ ord_less_eq_b @ F ) ) ).
% mono_onI
thf(fact_992_mono__onI,axiom,
! [A2: set_set_b,F: set_b > a] :
( ! [R2: set_b,S3: set_b] :
( ( member_set_b @ R2 @ A2 )
=> ( ( member_set_b @ S3 @ A2 )
=> ( ( ord_less_eq_set_b @ R2 @ S3 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_set_b_a @ A2 @ ord_less_eq_set_b @ ord_less_eq_a @ F ) ) ).
% mono_onI
thf(fact_993_mono__onD,axiom,
! [A2: set_a,F: a > a,R: a,S2: a] :
( ( monotone_on_a_a @ A2 @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( ord_less_eq_a @ R @ S2 )
=> ( ord_less_eq_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_994_mono__onD,axiom,
! [A2: set_a,F: a > nat,R: a,S2: a] :
( ( monotone_on_a_nat @ A2 @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( ord_less_eq_a @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_995_mono__onD,axiom,
! [A2: set_a,F: a > b,R: a,S2: a] :
( ( monotone_on_a_b @ A2 @ ord_less_eq_a @ ord_less_eq_b @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( ord_less_eq_a @ R @ S2 )
=> ( ord_less_eq_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_996_mono__onD,axiom,
! [A2: set_nat,F: nat > a,R: nat,S2: nat] :
( ( monotone_on_nat_a @ A2 @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_eq_nat @ R @ S2 )
=> ( ord_less_eq_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_997_mono__onD,axiom,
! [A2: set_nat,F: nat > nat,R: nat,S2: nat] :
( ( monotone_on_nat_nat @ A2 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_eq_nat @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_998_mono__onD,axiom,
! [A2: set_nat,F: nat > b,R: nat,S2: nat] :
( ( monotone_on_nat_b @ A2 @ ord_less_eq_nat @ ord_less_eq_b @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( ord_less_eq_nat @ R @ S2 )
=> ( ord_less_eq_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_999_mono__onD,axiom,
! [A2: set_b,F: b > a,R: b,S2: b] :
( ( monotone_on_b_a @ A2 @ ord_less_eq_b @ ord_less_eq_a @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( ord_less_eq_b @ R @ S2 )
=> ( ord_less_eq_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1000_mono__onD,axiom,
! [A2: set_b,F: b > nat,R: b,S2: b] :
( ( monotone_on_b_nat @ A2 @ ord_less_eq_b @ ord_less_eq_nat @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( ord_less_eq_b @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1001_mono__onD,axiom,
! [A2: set_b,F: b > b,R: b,S2: b] :
( ( monotone_on_b_b @ A2 @ ord_less_eq_b @ ord_less_eq_b @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( ord_less_eq_b @ R @ S2 )
=> ( ord_less_eq_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1002_mono__onD,axiom,
! [A2: set_set_b,F: set_b > a,R: set_b,S2: set_b] :
( ( monotone_on_set_b_a @ A2 @ ord_less_eq_set_b @ ord_less_eq_a @ F )
=> ( ( member_set_b @ R @ A2 )
=> ( ( member_set_b @ S2 @ A2 )
=> ( ( ord_less_eq_set_b @ R @ S2 )
=> ( ord_less_eq_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1003_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > a] :
( ( monotone_on_nat_a @ A2 @ Less_eq @ ord_less_eq_a @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1004_ord_Omono__on__def,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > a] :
( ( monotone_on_a_a @ A2 @ Less_eq @ ord_less_eq_a @ F )
= ( ! [R3: a,S4: a] :
( ( ( member_a @ R3 @ A2 )
& ( member_a @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1005_ord_Omono__on__def,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > a] :
( ( monotone_on_b_a @ A2 @ Less_eq @ ord_less_eq_a @ F )
= ( ! [R3: b,S4: b] :
( ( ( member_b @ R3 @ A2 )
& ( member_b @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1006_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1007_ord_Omono__on__def,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > nat] :
( ( monotone_on_a_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: a,S4: a] :
( ( ( member_a @ R3 @ A2 )
& ( member_a @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1008_ord_Omono__on__def,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > nat] :
( ( monotone_on_b_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: b,S4: b] :
( ( ( member_b @ R3 @ A2 )
& ( member_b @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1009_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > b] :
( ( monotone_on_nat_b @ A2 @ Less_eq @ ord_less_eq_b @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_b @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1010_ord_Omono__on__def,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > b] :
( ( monotone_on_b_b @ A2 @ Less_eq @ ord_less_eq_b @ F )
= ( ! [R3: b,S4: b] :
( ( ( member_b @ R3 @ A2 )
& ( member_b @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_b @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1011_ord_Omono__on__def,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > b] :
( ( monotone_on_a_b @ A2 @ Less_eq @ ord_less_eq_b @ F )
= ( ! [R3: a,S4: a] :
( ( ( member_a @ R3 @ A2 )
& ( member_a @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_b @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1012_ord_Omono__on__def,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_b] :
( ( monoto723715500276691686_set_b @ A2 @ Less_eq @ ord_less_eq_set_b @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A2 )
& ( member_nat @ S4 @ A2 )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_set_b @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1013_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > a] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_a @ A2 @ Less_eq @ ord_less_eq_a @ F ) ) ).
% ord.mono_onI
thf(fact_1014_ord_Omono__onI,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > a] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_a @ A2 @ Less_eq @ ord_less_eq_a @ F ) ) ).
% ord.mono_onI
thf(fact_1015_ord_Omono__onI,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > a] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_a @ A2 @ Less_eq @ ord_less_eq_a @ F ) ) ).
% ord.mono_onI
thf(fact_1016_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_1017_ord_Omono__onI,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > nat] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_1018_ord_Omono__onI,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > nat] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_1019_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > b] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_b @ A2 @ Less_eq @ ord_less_eq_b @ F ) ) ).
% ord.mono_onI
thf(fact_1020_ord_Omono__onI,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > b] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_b @ A2 @ Less_eq @ ord_less_eq_b @ F ) ) ).
% ord.mono_onI
thf(fact_1021_ord_Omono__onI,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > b] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_b @ A2 @ Less_eq @ ord_less_eq_b @ F ) ) ).
% ord.mono_onI
thf(fact_1022_ord_Omono__onI,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_b] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_set_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto723715500276691686_set_b @ A2 @ Less_eq @ ord_less_eq_set_b @ F ) ) ).
% ord.mono_onI
thf(fact_1023_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > a,R: nat,S2: nat] :
( ( monotone_on_nat_a @ A2 @ Less_eq @ ord_less_eq_a @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1024_ord_Omono__onD,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > a,R: a,S2: a] :
( ( monotone_on_a_a @ A2 @ Less_eq @ ord_less_eq_a @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1025_ord_Omono__onD,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > a,R: b,S2: b] :
( ( monotone_on_b_a @ A2 @ Less_eq @ ord_less_eq_a @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1026_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R: nat,S2: nat] :
( ( monotone_on_nat_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1027_ord_Omono__onD,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > nat,R: a,S2: a] :
( ( monotone_on_a_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1028_ord_Omono__onD,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > nat,R: b,S2: b] :
( ( monotone_on_b_nat @ A2 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1029_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > b,R: nat,S2: nat] :
( ( monotone_on_nat_b @ A2 @ Less_eq @ ord_less_eq_b @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1030_ord_Omono__onD,axiom,
! [A2: set_b,Less_eq: b > b > $o,F: b > b,R: b,S2: b] :
( ( monotone_on_b_b @ A2 @ Less_eq @ ord_less_eq_b @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1031_ord_Omono__onD,axiom,
! [A2: set_a,Less_eq: a > a > $o,F: a > b,R: a,S2: a] :
( ( monotone_on_a_b @ A2 @ Less_eq @ ord_less_eq_b @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1032_ord_Omono__onD,axiom,
! [A2: set_nat,Less_eq: nat > nat > $o,F: nat > set_b,R: nat,S2: nat] :
( ( monoto723715500276691686_set_b @ A2 @ Less_eq @ ord_less_eq_set_b @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_set_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1033_ord_Ostrict__mono__onD,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > a,R: nat,S2: nat] :
( ( monotone_on_nat_a @ A2 @ Less @ ord_less_a @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less @ R @ S2 )
=> ( ord_less_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1034_ord_Ostrict__mono__onD,axiom,
! [A2: set_a,Less: a > a > $o,F: a > a,R: a,S2: a] :
( ( monotone_on_a_a @ A2 @ Less @ ord_less_a @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( Less @ R @ S2 )
=> ( ord_less_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1035_ord_Ostrict__mono__onD,axiom,
! [A2: set_b,Less: b > b > $o,F: b > a,R: b,S2: b] :
( ( monotone_on_b_a @ A2 @ Less @ ord_less_a @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( Less @ R @ S2 )
=> ( ord_less_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1036_ord_Ostrict__mono__onD,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > b,R: nat,S2: nat] :
( ( monotone_on_nat_b @ A2 @ Less @ ord_less_b @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less @ R @ S2 )
=> ( ord_less_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1037_ord_Ostrict__mono__onD,axiom,
! [A2: set_b,Less: b > b > $o,F: b > b,R: b,S2: b] :
( ( monotone_on_b_b @ A2 @ Less @ ord_less_b @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( Less @ R @ S2 )
=> ( ord_less_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1038_ord_Ostrict__mono__onD,axiom,
! [A2: set_a,Less: a > a > $o,F: a > b,R: a,S2: a] :
( ( monotone_on_a_b @ A2 @ Less @ ord_less_b @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( Less @ R @ S2 )
=> ( ord_less_b @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1039_ord_Ostrict__mono__onD,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > nat,R: nat,S2: nat] :
( ( monotone_on_nat_nat @ A2 @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A2 )
=> ( ( member_nat @ S2 @ A2 )
=> ( ( Less @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1040_ord_Ostrict__mono__onD,axiom,
! [A2: set_a,Less: a > a > $o,F: a > nat,R: a,S2: a] :
( ( monotone_on_a_nat @ A2 @ Less @ ord_less_nat @ F )
=> ( ( member_a @ R @ A2 )
=> ( ( member_a @ S2 @ A2 )
=> ( ( Less @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1041_ord_Ostrict__mono__onD,axiom,
! [A2: set_b,Less: b > b > $o,F: b > nat,R: b,S2: b] :
( ( monotone_on_b_nat @ A2 @ Less @ ord_less_nat @ F )
=> ( ( member_b @ R @ A2 )
=> ( ( member_b @ S2 @ A2 )
=> ( ( Less @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1042_ord_Ostrict__mono__onD,axiom,
! [A2: set_set_nat,Less: set_nat > set_nat > $o,F: set_nat > a,R: set_nat,S2: set_nat] :
( ( monoto2395835772568396751_nat_a @ A2 @ Less @ ord_less_a @ F )
=> ( ( member_set_nat @ R @ A2 )
=> ( ( member_set_nat @ S2 @ A2 )
=> ( ( Less @ R @ S2 )
=> ( ord_less_a @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1043_ord_Ostrict__mono__onI,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > a] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_a @ A2 @ Less @ ord_less_a @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1044_ord_Ostrict__mono__onI,axiom,
! [A2: set_a,Less: a > a > $o,F: a > a] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_a @ A2 @ Less @ ord_less_a @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1045_ord_Ostrict__mono__onI,axiom,
! [A2: set_b,Less: b > b > $o,F: b > a] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_a @ A2 @ Less @ ord_less_a @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1046_ord_Ostrict__mono__onI,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > b] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_b @ A2 @ Less @ ord_less_b @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1047_ord_Ostrict__mono__onI,axiom,
! [A2: set_b,Less: b > b > $o,F: b > b] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_b @ A2 @ Less @ ord_less_b @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1048_ord_Ostrict__mono__onI,axiom,
! [A2: set_a,Less: a > a > $o,F: a > b] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_b @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_b @ A2 @ Less @ ord_less_b @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1049_ord_Ostrict__mono__onI,axiom,
! [A2: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A2 )
=> ( ( member_nat @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A2 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1050_ord_Ostrict__mono__onI,axiom,
! [A2: set_a,Less: a > a > $o,F: a > nat] :
( ! [R2: a,S3: a] :
( ( member_a @ R2 @ A2 )
=> ( ( member_a @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_a_nat @ A2 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1051_ord_Ostrict__mono__onI,axiom,
! [A2: set_b,Less: b > b > $o,F: b > nat] :
( ! [R2: b,S3: b] :
( ( member_b @ R2 @ A2 )
=> ( ( member_b @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_b_nat @ A2 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1052_ord_Ostrict__mono__onI,axiom,
! [A2: set_set_nat,Less: set_nat > set_nat > $o,F: set_nat > a] :
( ! [R2: set_nat,S3: set_nat] :
( ( member_set_nat @ R2 @ A2 )
=> ( ( member_set_nat @ S3 @ A2 )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_a @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto2395835772568396751_nat_a @ A2 @ Less @ ord_less_a @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1053_complete__interval,axiom,
! [A: a,B: a,P: a > $o] :
( ( ord_less_a @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C4: a] :
( ( ord_less_eq_a @ A @ C4 )
& ( ord_less_eq_a @ C4 @ B )
& ! [X4: a] :
( ( ( ord_less_eq_a @ A @ X4 )
& ( ord_less_a @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D2: a] :
( ! [X: a] :
( ( ( ord_less_eq_a @ A @ X )
& ( ord_less_a @ X @ D2 ) )
=> ( P @ X ) )
=> ( ord_less_eq_a @ D2 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1054_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D2: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ D2 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D2 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1055_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_1056_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1057_verit__comp__simplify1_I3_J,axiom,
! [B6: b,A6: b] :
( ( ~ ( ord_less_eq_b @ B6 @ A6 ) )
= ( ord_less_b @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1058_pinf_I6_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ~ ( ord_less_eq_a @ X4 @ T3 ) ) ).
% pinf(6)
thf(fact_1059_pinf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T3 ) ) ).
% pinf(6)
thf(fact_1060_pinf_I6_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ Z3 @ X4 )
=> ~ ( ord_less_eq_b @ X4 @ T3 ) ) ).
% pinf(6)
thf(fact_1061_pinf_I8_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ord_less_eq_a @ T3 @ X4 ) ) ).
% pinf(8)
thf(fact_1062_pinf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ T3 @ X4 ) ) ).
% pinf(8)
thf(fact_1063_pinf_I8_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ Z3 @ X4 )
=> ( ord_less_eq_b @ T3 @ X4 ) ) ).
% pinf(8)
thf(fact_1064_minf_I6_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ord_less_eq_a @ X4 @ T3 ) ) ).
% minf(6)
thf(fact_1065_minf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ T3 ) ) ).
% minf(6)
thf(fact_1066_minf_I6_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ X4 @ Z3 )
=> ( ord_less_eq_b @ X4 @ T3 ) ) ).
% minf(6)
thf(fact_1067_minf_I8_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ~ ( ord_less_eq_a @ T3 @ X4 ) ) ).
% minf(8)
thf(fact_1068_minf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ T3 @ X4 ) ) ).
% minf(8)
thf(fact_1069_minf_I8_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ X4 @ Z3 )
=> ~ ( ord_less_eq_b @ T3 @ X4 ) ) ).
% minf(8)
thf(fact_1070_idempotent__imp__retraction,axiom,
! [S: set_b,F: b > b] :
( ( topolo175937460483079870on_b_b @ S @ F )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ F @ S ) @ S )
=> ( ! [X: b] :
( ( member_b @ X @ S )
=> ( ( F @ ( F @ X ) )
= ( F @ X ) ) )
=> ( abstra5157962118735104162tion_b @ S @ ( image_b_b @ F @ S ) @ F ) ) ) ) ).
% idempotent_imp_retraction
thf(fact_1071_idempotent__imp__retraction,axiom,
! [S: set_nat,F: nat > nat] :
( ( topolo1182047505939668768at_nat @ S @ F )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ S ) @ S )
=> ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ( F @ ( F @ X ) )
= ( F @ X ) ) )
=> ( abstra7171991951520340845on_nat @ S @ ( image_nat_nat @ F @ S ) @ F ) ) ) ) ).
% idempotent_imp_retraction
thf(fact_1072_idempotent__imp__retraction,axiom,
! [S: set_a,F: a > a] :
( ( topolo2963393042455755902on_a_a @ S @ F )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ S ) @ S )
=> ( ! [X: a] :
( ( member_a @ X @ S )
=> ( ( F @ ( F @ X ) )
= ( F @ X ) ) )
=> ( abstra5157962118735104161tion_a @ S @ ( image_a_a @ F @ S ) @ F ) ) ) ) ).
% idempotent_imp_retraction
thf(fact_1073_retraction__idempotent,axiom,
! [S: set_nat,T2: set_nat,R: nat > nat,X2: nat] :
( ( abstra7171991951520340845on_nat @ S @ T2 @ R )
=> ( ( member_nat @ X2 @ S )
=> ( ( R @ ( R @ X2 ) )
= ( R @ X2 ) ) ) ) ).
% retraction_idempotent
thf(fact_1074_retraction__idempotent,axiom,
! [S: set_a,T2: set_a,R: a > a,X2: a] :
( ( abstra5157962118735104161tion_a @ S @ T2 @ R )
=> ( ( member_a @ X2 @ S )
=> ( ( R @ ( R @ X2 ) )
= ( R @ X2 ) ) ) ) ).
% retraction_idempotent
thf(fact_1075_retraction__idempotent,axiom,
! [S: set_b,T2: set_b,R: b > b,X2: b] :
( ( abstra5157962118735104162tion_b @ S @ T2 @ R )
=> ( ( member_b @ X2 @ S )
=> ( ( R @ ( R @ X2 ) )
= ( R @ X2 ) ) ) ) ).
% retraction_idempotent
thf(fact_1076_retraction__subset,axiom,
! [S: set_b,T2: set_b,R: b > b,S5: set_b] :
( ( abstra5157962118735104162tion_b @ S @ T2 @ R )
=> ( ( ord_less_eq_set_b @ T2 @ S5 )
=> ( ( ord_less_eq_set_b @ S5 @ S )
=> ( abstra5157962118735104162tion_b @ S5 @ T2 @ R ) ) ) ) ).
% retraction_subset
thf(fact_1077_retraction__subset,axiom,
! [S: set_nat,T2: set_nat,R: nat > nat,S5: set_nat] :
( ( abstra7171991951520340845on_nat @ S @ T2 @ R )
=> ( ( ord_less_eq_set_nat @ T2 @ S5 )
=> ( ( ord_less_eq_set_nat @ S5 @ S )
=> ( abstra7171991951520340845on_nat @ S5 @ T2 @ R ) ) ) ) ).
% retraction_subset
thf(fact_1078_retraction__subset,axiom,
! [S: set_a,T2: set_a,R: a > a,S5: set_a] :
( ( abstra5157962118735104161tion_a @ S @ T2 @ R )
=> ( ( ord_less_eq_set_a @ T2 @ S5 )
=> ( ( ord_less_eq_set_a @ S5 @ S )
=> ( abstra5157962118735104161tion_a @ S5 @ T2 @ R ) ) ) ) ).
% retraction_subset
thf(fact_1079_verit__comp__simplify1_I2_J,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1080_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1081_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1082_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1083_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1084_verit__comp__simplify1_I2_J,axiom,
! [A: b] : ( ord_less_eq_b @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1085_verit__comp__simplify1_I2_J,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1086_verit__comp__simplify1_I2_J,axiom,
! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1087_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1088_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1089_verit__la__disequality,axiom,
! [A: b,B: b] :
( ( A = B )
| ~ ( ord_less_eq_b @ A @ B )
| ~ ( ord_less_eq_b @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1090_ex__gt__or__lt,axiom,
! [A: a] :
? [B4: a] :
( ( ord_less_a @ A @ B4 )
| ( ord_less_a @ B4 @ A ) ) ).
% ex_gt_or_lt
thf(fact_1091_verit__comp__simplify1_I1_J,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1092_verit__comp__simplify1_I1_J,axiom,
! [A: b] :
~ ( ord_less_b @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1093_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1094_pinf_I1_J,axiom,
! [P: a > $o,P4: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1095_pinf_I1_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X: b] :
( ( ord_less_b @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: b] :
! [X: b] :
( ( ord_less_b @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1096_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1097_pinf_I2_J,axiom,
! [P: a > $o,P4: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1098_pinf_I2_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X: b] :
( ( ord_less_b @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: b] :
! [X: b] :
( ( ord_less_b @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1099_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1100_pinf_I3_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(3)
thf(fact_1101_pinf_I3_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ Z3 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(3)
thf(fact_1102_pinf_I3_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(3)
thf(fact_1103_pinf_I4_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(4)
thf(fact_1104_pinf_I4_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ Z3 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(4)
thf(fact_1105_pinf_I4_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(4)
thf(fact_1106_pinf_I5_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ~ ( ord_less_a @ X4 @ T3 ) ) ).
% pinf(5)
thf(fact_1107_pinf_I5_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ Z3 @ X4 )
=> ~ ( ord_less_b @ X4 @ T3 ) ) ).
% pinf(5)
thf(fact_1108_pinf_I5_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T3 ) ) ).
% pinf(5)
thf(fact_1109_pinf_I7_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ Z3 @ X4 )
=> ( ord_less_a @ T3 @ X4 ) ) ).
% pinf(7)
thf(fact_1110_pinf_I7_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ Z3 @ X4 )
=> ( ord_less_b @ T3 @ X4 ) ) ).
% pinf(7)
thf(fact_1111_pinf_I7_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_nat @ T3 @ X4 ) ) ).
% pinf(7)
thf(fact_1112_minf_I1_J,axiom,
! [P: a > $o,P4: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1113_minf_I1_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X: b] :
( ( ord_less_b @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: b] :
! [X: b] :
( ( ord_less_b @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1114_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1115_minf_I2_J,axiom,
! [P: a > $o,P4: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: a] :
! [X: a] :
( ( ord_less_a @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1116_minf_I2_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X: b] :
( ( ord_less_b @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: b] :
! [X: b] :
( ( ord_less_b @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1117_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1118_minf_I3_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( X4 != T3 ) ) ).
% minf(3)
thf(fact_1119_minf_I3_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ X4 @ Z3 )
=> ( X4 != T3 ) ) ).
% minf(3)
thf(fact_1120_minf_I3_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T3 ) ) ).
% minf(3)
thf(fact_1121_minf_I4_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( X4 != T3 ) ) ).
% minf(4)
thf(fact_1122_minf_I4_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ X4 @ Z3 )
=> ( X4 != T3 ) ) ).
% minf(4)
thf(fact_1123_minf_I4_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T3 ) ) ).
% minf(4)
thf(fact_1124_minf_I5_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ( ord_less_a @ X4 @ T3 ) ) ).
% minf(5)
thf(fact_1125_minf_I5_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ X4 @ Z3 )
=> ( ord_less_b @ X4 @ T3 ) ) ).
% minf(5)
thf(fact_1126_minf_I5_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_nat @ X4 @ T3 ) ) ).
% minf(5)
thf(fact_1127_minf_I7_J,axiom,
! [T3: a] :
? [Z3: a] :
! [X4: a] :
( ( ord_less_a @ X4 @ Z3 )
=> ~ ( ord_less_a @ T3 @ X4 ) ) ).
% minf(7)
thf(fact_1128_minf_I7_J,axiom,
! [T3: b] :
? [Z3: b] :
! [X4: b] :
( ( ord_less_b @ X4 @ Z3 )
=> ~ ( ord_less_b @ T3 @ X4 ) ) ).
% minf(7)
thf(fact_1129_minf_I7_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_nat @ T3 @ X4 ) ) ).
% minf(7)
thf(fact_1130_retraction,axiom,
( abstra5157962118735104162tion_b
= ( ^ [S6: set_b,T4: set_b,R3: b > b] :
( ( ord_less_eq_set_b @ T4 @ S6 )
& ( topolo175937460483079870on_b_b @ S6 @ R3 )
& ( ( image_b_b @ R3 @ S6 )
= T4 )
& ! [X3: b] :
( ( member_b @ X3 @ T4 )
=> ( ( R3 @ X3 )
= X3 ) ) ) ) ) ).
% retraction
thf(fact_1131_retraction,axiom,
( abstra7171991951520340845on_nat
= ( ^ [S6: set_nat,T4: set_nat,R3: nat > nat] :
( ( ord_less_eq_set_nat @ T4 @ S6 )
& ( topolo1182047505939668768at_nat @ S6 @ R3 )
& ( ( image_nat_nat @ R3 @ S6 )
= T4 )
& ! [X3: nat] :
( ( member_nat @ X3 @ T4 )
=> ( ( R3 @ X3 )
= X3 ) ) ) ) ) ).
% retraction
thf(fact_1132_retraction,axiom,
( abstra5157962118735104161tion_a
= ( ^ [S6: set_a,T4: set_a,R3: a > a] :
( ( ord_less_eq_set_a @ T4 @ S6 )
& ( topolo2963393042455755902on_a_a @ S6 @ R3 )
& ( ( image_a_a @ R3 @ S6 )
= T4 )
& ! [X3: a] :
( ( member_a @ X3 @ T4 )
=> ( ( R3 @ X3 )
= X3 ) ) ) ) ) ).
% retraction
thf(fact_1133_retractionE,axiom,
! [S: set_b,T2: set_b,R: b > b] :
( ( abstra5157962118735104162tion_b @ S @ T2 @ R )
=> ~ ( ( T2
= ( image_b_b @ R @ S ) )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ R @ S ) @ S )
=> ( ( topolo175937460483079870on_b_b @ S @ R )
=> ~ ! [X4: b] :
( ( member_b @ X4 @ S )
=> ( ( R @ ( R @ X4 ) )
= ( R @ X4 ) ) ) ) ) ) ) ).
% retractionE
thf(fact_1134_retractionE,axiom,
! [S: set_nat,T2: set_nat,R: nat > nat] :
( ( abstra7171991951520340845on_nat @ S @ T2 @ R )
=> ~ ( ( T2
= ( image_nat_nat @ R @ S ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ S ) @ S )
=> ( ( topolo1182047505939668768at_nat @ S @ R )
=> ~ ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ( ( R @ ( R @ X4 ) )
= ( R @ X4 ) ) ) ) ) ) ) ).
% retractionE
thf(fact_1135_retractionE,axiom,
! [S: set_a,T2: set_a,R: a > a] :
( ( abstra5157962118735104161tion_a @ S @ T2 @ R )
=> ~ ( ( T2
= ( image_a_a @ R @ S ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R @ S ) @ S )
=> ( ( topolo2963393042455755902on_a_a @ S @ R )
=> ~ ! [X4: a] :
( ( member_a @ X4 @ S )
=> ( ( R @ ( R @ X4 ) )
= ( R @ X4 ) ) ) ) ) ) ) ).
% retractionE
thf(fact_1136_retraction__def,axiom,
( abstra5157962118735104162tion_b
= ( ^ [S6: set_b,T4: set_b,R3: b > b] :
( ( ord_less_eq_set_b @ T4 @ S6 )
& ( topolo175937460483079870on_b_b @ S6 @ R3 )
& ( ord_less_eq_set_b @ ( image_b_b @ R3 @ S6 ) @ T4 )
& ! [X3: b] :
( ( member_b @ X3 @ T4 )
=> ( ( R3 @ X3 )
= X3 ) ) ) ) ) ).
% retraction_def
thf(fact_1137_retraction__def,axiom,
( abstra7171991951520340845on_nat
= ( ^ [S6: set_nat,T4: set_nat,R3: nat > nat] :
( ( ord_less_eq_set_nat @ T4 @ S6 )
& ( topolo1182047505939668768at_nat @ S6 @ R3 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ R3 @ S6 ) @ T4 )
& ! [X3: nat] :
( ( member_nat @ X3 @ T4 )
=> ( ( R3 @ X3 )
= X3 ) ) ) ) ) ).
% retraction_def
thf(fact_1138_retraction__def,axiom,
( abstra5157962118735104161tion_a
= ( ^ [S6: set_a,T4: set_a,R3: a > a] :
( ( ord_less_eq_set_a @ T4 @ S6 )
& ( topolo2963393042455755902on_a_a @ S6 @ R3 )
& ( ord_less_eq_set_a @ ( image_a_a @ R3 @ S6 ) @ T4 )
& ! [X3: a] :
( ( member_a @ X3 @ T4 )
=> ( ( R3 @ X3 )
= X3 ) ) ) ) ) ).
% retraction_def
thf(fact_1139_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or5939364468397584554Than_a @ A @ B ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_1140_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ A @ B ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_1141_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A @ B ) @ ( set_or2348907005316661231_set_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_set_a @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1142_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat,D: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or9137876137106135879et_nat @ A @ B ) @ ( set_or5410080298493297259et_nat @ C @ D ) )
= ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ C @ A )
& ( ord_less_set_set_nat @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1143_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_set_b,B: set_set_b,C: set_set_b,D: set_set_b] :
( ( ord_le3201067847557142847_set_b @ ( set_or4832498528752608884_set_b @ A @ B ) @ ( set_or7788675546676248656_set_b @ C @ D ) )
= ( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( ord_le3795704787696855135_set_b @ C @ A )
& ( ord_less_set_set_b @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1144_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: b,B: b,C: b,D: b] :
( ( ord_less_eq_set_b @ ( set_or672772299803893940Most_b @ A @ B ) @ ( set_or5139330845457685136Than_b @ C @ D ) )
= ( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_b @ C @ A )
& ( ord_less_b @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1145_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A @ B ) @ ( set_or5139330845457685135Than_a @ C @ D ) )
= ( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ A )
& ( ord_less_a @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1146_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or3540276404033026485et_nat @ C @ D ) )
= ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_set_nat @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1147_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_b,B: set_b,C: set_b,D: set_b] :
( ( ord_le3795704787696855135_set_b @ ( set_or6288561114688587156_set_b @ A @ B ) @ ( set_or2348907009619890032_set_b @ C @ D ) )
= ( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ C @ A )
& ( ord_less_set_b @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1148_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_nat @ B @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1149_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ A @ B ) @ ( set_or672772299803893939Most_a @ C @ D ) )
= ( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_1150_Compl__subset__Compl__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ ( uminus_uminus_set_b @ A2 ) @ ( uminus_uminus_set_b @ B2 ) )
= ( ord_less_eq_set_b @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1151_Compl__subset__Compl__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) )
= ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1152_Compl__subset__Compl__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B2 ) )
= ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1153_Compl__subset__Compl__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ A2 ) @ ( uminus613421341184616069et_nat @ B2 ) )
= ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1154_Compl__subset__Compl__iff,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ ( uminus6174936397961129654_set_b @ A2 ) @ ( uminus6174936397961129654_set_b @ B2 ) )
= ( ord_le3795704787696855135_set_b @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1155_Compl__anti__mono,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ ( uminus_uminus_set_b @ B2 ) @ ( uminus_uminus_set_b @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_1156_Compl__anti__mono,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B2 ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_1157_Compl__anti__mono,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B2 ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_1158_Compl__anti__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ B2 ) @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_1159_Compl__anti__mono,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ( ord_le3795704787696855135_set_b @ ( uminus6174936397961129654_set_b @ B2 ) @ ( uminus6174936397961129654_set_b @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_1160_ivl__diff,axiom,
! [I: a,N3: a,M3: a] :
( ( ord_less_eq_a @ I @ N3 )
=> ( ( minus_minus_set_a @ ( set_or5139330845457685135Than_a @ I @ M3 ) @ ( set_or5139330845457685135Than_a @ I @ N3 ) )
= ( set_or5139330845457685135Than_a @ N3 @ M3 ) ) ) ).
% ivl_diff
thf(fact_1161_ivl__diff,axiom,
! [I: b,N3: b,M3: b] :
( ( ord_less_eq_b @ I @ N3 )
=> ( ( minus_minus_set_b @ ( set_or5139330845457685136Than_b @ I @ M3 ) @ ( set_or5139330845457685136Than_b @ I @ N3 ) )
= ( set_or5139330845457685136Than_b @ N3 @ M3 ) ) ) ).
% ivl_diff
thf(fact_1162_ivl__diff,axiom,
! [I: nat,N3: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M3 ) @ ( set_or4665077453230672383an_nat @ I @ N3 ) )
= ( set_or4665077453230672383an_nat @ N3 @ M3 ) ) ) ).
% ivl_diff
thf(fact_1163_greaterThanLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or8625682525731655386et_nat @ L @ U ) )
= ( ( ord_less_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_1164_greaterThanLessThan__iff,axiom,
! [I: set_b,L: set_b,U: set_b] :
( ( member_set_b @ I @ ( set_or6017932781039335819_set_b @ L @ U ) )
= ( ( ord_less_set_b @ L @ I )
& ( ord_less_set_b @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_1165_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_1166_greaterThanLessThan__iff,axiom,
! [I: b,L: b,U: b] :
( ( member_b @ I @ ( set_or5939364468397584555Than_b @ L @ U ) )
= ( ( ord_less_b @ L @ I )
& ( ord_less_b @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_1167_greaterThanLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_1168_atLeastLessThan__iff,axiom,
! [I: set_b,L: set_b,U: set_b] :
( ( member_set_b @ I @ ( set_or2348907009619890032_set_b @ L @ U ) )
= ( ( ord_less_eq_set_b @ L @ I )
& ( ord_less_set_b @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1169_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_1170_atLeastLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1171_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_1172_atLeastLessThan__iff,axiom,
! [I: b,L: b,U: b] :
( ( member_b @ I @ ( set_or5139330845457685136Than_b @ L @ U ) )
= ( ( ord_less_eq_b @ L @ I )
& ( ord_less_b @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1173_atLeastLessThan__iff,axiom,
! [I: set_set_nat,L: set_set_nat,U: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or5410080298493297259et_nat @ L @ U ) )
= ( ( ord_le6893508408891458716et_nat @ L @ I )
& ( ord_less_set_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1174_atLeastLessThan__iff,axiom,
! [I: set_set_b,L: set_set_b,U: set_set_b] :
( ( member_set_set_b @ I @ ( set_or7788675546676248656_set_b @ L @ U ) )
= ( ( ord_le3795704787696855135_set_b @ L @ I )
& ( ord_less_set_set_b @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1175_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1176_ivl__subset,axiom,
! [I: b,J: b,M3: b,N3: b] :
( ( ord_less_eq_set_b @ ( set_or5139330845457685136Than_b @ I @ J ) @ ( set_or5139330845457685136Than_b @ M3 @ N3 ) )
= ( ( ord_less_eq_b @ J @ I )
| ( ( ord_less_eq_b @ M3 @ I )
& ( ord_less_eq_b @ J @ N3 ) ) ) ) ).
% ivl_subset
thf(fact_1177_ivl__subset,axiom,
! [I: a,J: a,M3: a,N3: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ I @ J ) @ ( set_or5139330845457685135Than_a @ M3 @ N3 ) )
= ( ( ord_less_eq_a @ J @ I )
| ( ( ord_less_eq_a @ M3 @ I )
& ( ord_less_eq_a @ J @ N3 ) ) ) ) ).
% ivl_subset
thf(fact_1178_ivl__subset,axiom,
! [I: nat,J: nat,M3: nat,N3: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M3 @ N3 ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M3 @ I )
& ( ord_less_eq_nat @ J @ N3 ) ) ) ) ).
% ivl_subset
thf(fact_1179_greaterThanAtMost__iff,axiom,
! [I: set_b,L: set_b,U: set_b] :
( ( member_set_b @ I @ ( set_or2503527073787596079_set_b @ L @ U ) )
= ( ( ord_less_set_b @ L @ I )
& ( ord_less_eq_set_b @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1180_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_1181_greaterThanAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or7074010630789208630et_nat @ L @ U ) )
= ( ( ord_less_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1182_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_1183_greaterThanAtMost__iff,axiom,
! [I: b,L: b,U: b] :
( ( member_b @ I @ ( set_or4472690218693186639Most_b @ L @ U ) )
= ( ( ord_less_b @ L @ I )
& ( ord_less_eq_b @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1184_greaterThanAtMost__iff,axiom,
! [I: set_set_nat,L: set_set_nat,U: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or7489957300529979116et_nat @ L @ U ) )
= ( ( ord_less_set_set_nat @ L @ I )
& ( ord_le6893508408891458716et_nat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1185_greaterThanAtMost__iff,axiom,
! [I: set_set_b,L: set_set_b,U: set_set_b] :
( ( member_set_set_b @ I @ ( set_or6144489860636312079_set_b @ L @ U ) )
= ( ( ord_less_set_set_b @ L @ I )
& ( ord_le3795704787696855135_set_b @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1186_greaterThanAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1187_atLeastLessThan__eq__iff,axiom,
! [A: b,B: b,C: b,D: b] :
( ( ord_less_b @ A @ B )
=> ( ( ord_less_b @ C @ D )
=> ( ( ( set_or5139330845457685136Than_b @ A @ B )
= ( set_or5139330845457685136Than_b @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_1188_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_1189_diff__diff__cancel,axiom,
! [I: nat,N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1190_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_1191_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_1192_Suc__diff__diff,axiom,
! [M3: nat,N3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N3 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N3 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1193_diff__Suc__Suc,axiom,
! [M3: nat,N3: nat] :
( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N3 ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ).
% diff_Suc_Suc
thf(fact_1194_Suc__diff__le,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_eq_nat @ N3 @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N3 )
= ( suc @ ( minus_minus_nat @ M3 @ N3 ) ) ) ) ).
% Suc_diff_le
thf(fact_1195_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( set_or1210151606488870762an_nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_1196_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_1197_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_1198_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_1199_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_1200_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1201_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1202_Suc__diff__Suc,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ N3 @ M3 )
=> ( ( suc @ ( minus_minus_nat @ M3 @ ( suc @ N3 ) ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ) ).
% Suc_diff_Suc
thf(fact_1203_diff__less__Suc,axiom,
! [M3: nat,N3: nat] : ( ord_less_nat @ ( minus_minus_nat @ M3 @ N3 ) @ ( suc @ M3 ) ) ).
% diff_less_Suc
thf(fact_1204_diff__less__mono2,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_1205_less__imp__diff__less,axiom,
! [J: nat,K: nat,N3: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N3 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1206_diff__le__mono2,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1207_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1208_diff__le__self,axiom,
! [M3: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N3 ) @ M3 ) ).
% diff_le_self
thf(fact_1209_diff__le__mono,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_1210_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1211_le__diff__iff,axiom,
! [K: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_1212_eq__diff__iff,axiom,
! [K: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N3 @ K ) )
= ( M3 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_1213_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1214_less__diff__iff,axiom,
! [K: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_nat @ M3 @ N3 ) ) ) ) ).
% less_diff_iff
thf(fact_1215_atLeastLessThan__singleton,axiom,
! [M3: nat] :
( ( set_or4665077453230672383an_nat @ M3 @ ( suc @ M3 ) )
= ( insert_nat @ M3 @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_1216_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_1217_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_1218_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_1219_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% atLeast_Suc
thf(fact_1220_Icc__eq__insert__lb__nat,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( set_or1269000886237332187st_nat @ M3 @ N3 )
= ( insert_nat @ M3 @ ( set_or1269000886237332187st_nat @ ( suc @ M3 ) @ N3 ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_1221_atLeastAtMostSuc__conv,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) )
=> ( ( set_or1269000886237332187st_nat @ M3 @ ( suc @ N3 ) )
= ( insert_nat @ ( suc @ N3 ) @ ( set_or1269000886237332187st_nat @ M3 @ N3 ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_1222_atLeastAtMost__insertL,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( insert_nat @ M3 @ ( set_or1269000886237332187st_nat @ ( suc @ M3 ) @ N3 ) )
= ( set_or1269000886237332187st_nat @ M3 @ N3 ) ) ) ).
% atLeastAtMost_insertL
thf(fact_1223_atLeastLessThanSuc,axiom,
! [M3: nat,N3: nat] :
( ( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( set_or4665077453230672383an_nat @ M3 @ ( suc @ N3 ) )
= ( insert_nat @ N3 @ ( set_or4665077453230672383an_nat @ M3 @ N3 ) ) ) )
& ( ~ ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( set_or4665077453230672383an_nat @ M3 @ ( suc @ N3 ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_1224_atLeast1__lessThan__eq__remove0,axiom,
! [N3: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N3 )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N3 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_1225_atLeast1__atMost__eq__remove0,axiom,
! [N3: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N3 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_1226_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1227_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1228_diff__is__0__eq_H,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1229_diff__is__0__eq,axiom,
! [M3: nat,N3: nat] :
( ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_1230_zero__less__diff,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% zero_less_diff
thf(fact_1231_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1232_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_1233_Suc__pred,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
= N3 ) ) ).
% Suc_pred
thf(fact_1234_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1235_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_1236_diffs0__imp__equal,axiom,
! [M3: nat,N3: nat] :
( ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M3 )
= zero_zero_nat )
=> ( M3 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_1237_diff__less,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N3 ) @ M3 ) ) ) ).
% diff_less
thf(fact_1238_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_1239_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1240_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1241_lessThan__empty__iff,axiom,
! [N3: nat] :
( ( ( set_ord_lessThan_nat @ N3 )
= bot_bot_set_nat )
= ( N3 = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1242_atLeastLessThan0,axiom,
! [M3: nat] :
( ( set_or4665077453230672383an_nat @ M3 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1243_diff__Suc__less,axiom,
! [N3: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ ( minus_minus_nat @ N3 @ ( suc @ I ) ) @ N3 ) ) ).
% diff_Suc_less
thf(fact_1244_atLeast0__atMost__Suc,axiom,
! [N3: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) )
= ( insert_nat @ ( suc @ N3 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_1245_atLeast0__lessThan__Suc,axiom,
! [N3: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N3 ) )
= ( insert_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_1246_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_1247_lessThan__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N3 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_1248_atMost__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_ord_atMost_nat @ ( suc @ N3 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_1249_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N3 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_1250_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_1251_ex__nat__less__eq,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ( P @ M2 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1252_all__nat__less__eq,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1253_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1254_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_1255_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_1256_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_1257_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_1258_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_1259_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1260_bounded__nat__set__is__finite,axiom,
! [N: set_nat,N3: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ord_less_nat @ X @ N3 ) )
=> ( finite_finite_nat @ N ) ) ).
% bounded_nat_set_is_finite
thf(fact_1261_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1262_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1263_diff__mult__distrib2,axiom,
! [K: nat,M3: nat,N3: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M3 @ N3 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N3 ) ) ) ).
% diff_mult_distrib2
thf(fact_1264_diff__mult__distrib,axiom,
! [M3: nat,N3: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M3 @ N3 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1265_subset__eq__atLeast0__lessThan__finite,axiom,
! [N: set_nat,N3: nat] :
( ( ord_less_eq_set_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
=> ( finite_finite_nat @ N ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_1266_subset__eq__atLeast0__atMost__finite,axiom,
! [N: set_nat,N3: nat] :
( ( ord_less_eq_set_nat @ N @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
=> ( finite_finite_nat @ N ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1267_Suc__diff__1,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= N3 ) ) ).
% Suc_diff_1
thf(fact_1268_diff__Suc__1,axiom,
! [N3: nat] :
( ( minus_minus_nat @ ( suc @ N3 ) @ one_one_nat )
= N3 ) ).
% diff_Suc_1
thf(fact_1269_diff__Suc__eq__diff__pred,axiom,
! [M3: nat,N3: nat] :
( ( minus_minus_nat @ M3 @ ( suc @ N3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1270_Suc__pred_H,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( N3
= ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1271_Suc__diff__eq__diff__pred,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N3 )
= ( minus_minus_nat @ M3 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1272_image__Suc__lessThan,axiom,
! [N3: nat] :
( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N3 ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ).
% image_Suc_lessThan
thf(fact_1273_image__Suc__atMost,axiom,
! [N3: nat] :
( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N3 ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N3 ) ) ) ).
% image_Suc_atMost
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_set_b @ ( set_or672772299803893940Most_b @ ( f @ a2 ) @ ( f @ b2 ) ) @ ( image_a_b @ f @ ( set_or672772299803893939Most_a @ a2 @ b2 ) ) ).
%------------------------------------------------------------------------------