TPTP Problem File: SLH0633^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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 : Clique_and_Monotone_Circuits/0005_Clique_Large_Monotone_Circuits/prob_00097_002706__16087898_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1400 ( 597 unt; 120 typ; 0 def)
% Number of atoms : 3829 (1289 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 11682 ( 425 ~; 50 |; 265 &;9164 @)
% ( 0 <=>;1778 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 471 ( 471 >; 0 *; 0 +; 0 <<)
% Number of symbols : 116 ( 113 usr; 9 con; 0-5 aty)
% Number of variables : 3492 ( 230 ^;3167 !; 95 ?;3492 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:47:52.708
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_set_set_set_nat: $tType ).
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_I_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
set_set_nat_o: $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_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (113)
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions,type,
assump5453534214990993103ptions: nat > nat > nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Nat__Onat,type,
clique6722202388162463298od_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Set__Oset_It__Nat__Onat_J,type,
clique8906516429304539640et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060K_062,type,
clique3326749438856946062irst_K: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov,type,
clique5033774636164728513irst_v: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
comple3806919086088850358_nat_o: set_set_nat_o > set_nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Nat__Onat_J,type,
condit5477540289124974626et_nat: set_set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
condit3670481866171438296et_nat: set_set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite6739761609112101331et_nat: set_set_set_nat > $o ).
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_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
minus_2447799839930672331et_nat: set_set_set_nat > set_set_set_nat > set_set_set_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_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
inf_in5711780100303410308et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Lattices_Osemilattice__neutr_001t__Set__Oset_It__Nat__Onat_J,type,
semila1241773964035338532et_nat: ( set_nat > set_nat > set_nat ) > set_nat > $o ).
thf(sy_c_Lattices_Osemilattice__neutr_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
semila7398651959140203994et_nat: ( set_set_nat > set_set_nat > set_set_nat ) > set_set_nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
sup_sup_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Nat__Onat_J_M_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
sup_su3254969269353549134_nat_o: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > set_nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
sup_su4213647025997063966et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3014633134055518761et_nat: set_set_nat > set_nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
lattic700688560247204575et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3835124923745554447et_nat: set_set_nat > set_nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
lattic7928989940735914181et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
measur5248428813077667851et_nat: set_set_nat > ( set_nat > set_nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
measur496615480034414785et_nat: set_set_nat > ( set_nat > set_set_nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
measur6219391137901972417et_nat: set_set_set_nat > ( set_set_nat > set_nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
measur8782999752332551287et_nat: set_set_set_nat > ( set_set_nat > set_set_nat ) > $o ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
measur5257060982548205482et_nat: set_nat > set_nat > ( set_nat > set_nat ) > set_nat > set_nat > set_nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
measur6315251617118684256et_nat: set_nat > set_nat > ( set_nat > set_set_nat ) > set_nat > set_nat > set_nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
measur5433014825823400032et_nat: set_set_nat > set_set_nat > ( set_set_nat > set_nat ) > set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
measur7844494817220136214et_nat: set_set_nat > set_set_nat > ( set_set_nat > set_set_nat ) > set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_nat > $o ).
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_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_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__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le6539261115178940645et_nat: ( $o > set_set_nat ) > ( $o > set_set_nat ) > $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_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_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_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Nat__Onat_J,type,
pow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_4687162037615663680et_nat: ( ( set_nat > $o ) > set_set_nat ) > set_set_nat_o > set_set_set_nat ).
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_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
insert_set_set_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_It__Nat__Onat_J,type,
is_empty_set_nat: set_set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
is_singleton_set_nat: set_set_nat > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__Nat__Onat_J,type,
remove_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(sy_c_Sigma__Algebra_Oalgebra_001t__Nat__Onat,type,
sigma_algebra_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Oalgebra_001t__Set__Oset_It__Nat__Onat_J,type,
sigma_5697435980195335136et_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Oclosed__cdi_001t__Nat__Onat,type,
sigma_closed_cdi_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Oclosed__cdi_001t__Set__Oset_It__Nat__Onat_J,type,
sigma_3476109907249799476et_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Oring__of__sets_001t__Nat__Onat,type,
sigma_8325262026724180568ts_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Oring__of__sets_001t__Set__Oset_It__Nat__Onat_J,type,
sigma_5512005348123348494et_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001t__Nat__Onat,type,
sigma_8817008012692346403ra_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001t__Set__Oset_It__Nat__Onat_J,type,
sigma_3186688988589110745et_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001t__Nat__Onat,type,
sigma_sigma_sets_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001t__Set__Oset_It__Nat__Onat_J,type,
sigma_5025102979728185774et_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Sigma__Algebra_Osmallest__ccdi__sets_001t__Nat__Onat,type,
sigma_5553761350045521333ts_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Sigma__Algebra_Osmallest__ccdi__sets_001t__Set__Oset_It__Nat__Onat_J,type,
sigma_1895591208183295339et_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Sigma__Algebra_Osubset__class_001t__Nat__Onat,type,
sigma_9101811122110323416ss_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Osubset__class_001t__Set__Oset_It__Nat__Onat_J,type,
sigma_1454238057069045390et_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Zorn_Ochains_001t__Nat__Onat,type,
chains_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Zorn_Ochains_001t__Set__Oset_It__Nat__Onat_J,type,
chains_set_nat: set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Nat__Onat,type,
pred_chain_nat: set_nat > ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_It__Nat__Onat_J,type,
pred_chain_set_nat: set_set_nat > ( set_nat > set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
pred_c5700569349699901905et_nat: set_set_set_nat > ( set_set_nat > set_set_nat > $o ) > set_set_set_nat > $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_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member2946998982187404937et_nat: set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_v_G,type,
g: set_set_nat ).
thf(sy_v_e,type,
e: set_nat ).
thf(sy_v_x____,type,
x: nat ).
thf(sy_v_y____,type,
y: nat ).
% Relevant facts (1279)
thf(fact_0_xy,axiom,
x != y ).
% xy
thf(fact_1_assms_I2_J,axiom,
member_set_nat @ e @ g ).
% assms(2)
thf(fact_2_e,axiom,
( e
= ( insert_nat @ x @ ( insert_nat @ y @ bot_bot_set_nat ) ) ) ).
% e
thf(fact_3_v__mono,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_mono
thf(fact_4_v__union,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_union
thf(fact_5_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_6_first__assumptions_Ov__mono,axiom,
! [L: nat,P: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_7_first__assumptions_Ov__union,axiom,
! [L: nat,P: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_union
thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Ay_O_A_092_060lbrakk_062e_A_061_A_123x_M_Ay_125_059_Ax_A_092_060noteq_062_Ay_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: nat,Y: nat] :
( ( e
= ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
=> ( X = Y ) ) ).
% \<open>\<And>thesis. (\<And>x y. \<lbrakk>e = {x, y}; x \<noteq> y\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_9_singleton__insert__inj__eq,axiom,
! [B: set_nat,A: set_nat,A2: set_set_nat] :
( ( ( insert_set_nat @ B @ bot_bot_set_set_nat )
= ( insert_set_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_10_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_11_singleton__insert__inj__eq_H,axiom,
! [A: set_nat,A2: set_set_nat,B: set_nat] :
( ( ( insert_set_nat @ A @ A2 )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( ( A = B )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_12_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_13_Un__insert__left,axiom,
! [A: set_nat,B2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C )
= ( insert_set_nat @ A @ ( sup_sup_set_set_nat @ B2 @ C ) ) ) ).
% Un_insert_left
thf(fact_14_Un__insert__left,axiom,
! [A: nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B2 ) @ C )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% Un_insert_left
thf(fact_15_Un__insert__right,axiom,
! [A2: set_set_nat,A: set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( insert_set_nat @ A @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_16_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_17_Un__empty,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_18_Un__empty,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ( A2 = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% Un_empty
thf(fact_19_Un__subset__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ C )
= ( ( ord_le6893508408891458716et_nat @ A2 @ C )
& ( ord_le6893508408891458716et_nat @ B2 @ C ) ) ) ).
% Un_subset_iff
thf(fact_20_Un__subset__iff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( ( ord_less_eq_set_nat @ A2 @ C )
& ( ord_less_eq_set_nat @ B2 @ C ) ) ) ).
% Un_subset_iff
thf(fact_21_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_22_sup__bot__left,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_23_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_24_sup__bot__right,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ bot_bot_set_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_25_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_26_bot__eq__sup__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_27_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y2 )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_28_sup__eq__bot__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X2 @ Y2 )
= bot_bot_set_set_nat )
= ( ( X2 = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_29_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_30_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ( A = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_31_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_32_sup__bot_Oleft__neutral,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
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_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_35_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_36_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_37_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_38_empty__Collect__eq,axiom,
! [P2: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P2 ) )
= ( ! [X3: set_nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_39_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_40_Collect__empty__eq,axiom,
! [P2: set_nat > $o] :
( ( ( collect_set_nat @ P2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_41_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_42_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_43_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_44_empty__iff,axiom,
! [C2: set_nat] :
~ ( member_set_nat @ C2 @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_45_insert__absorb2,axiom,
! [X2: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A2 ) )
= ( insert_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_46_insert__absorb2,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ X2 @ ( insert_set_nat @ X2 @ A2 ) )
= ( insert_set_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_47_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_48_insert__iff,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_49_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_50_insertCI,axiom,
! [A: set_nat,B2: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_51_sup_Oright__idem,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ B )
= ( sup_sup_set_set_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_52_sup_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_53_sup__left__idem,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_54_sup__left__idem,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_55_sup_Oleft__idem,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_set_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_56_sup_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_57_sup__idem,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_58_sup__idem,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_59_sup_Oidem,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_60_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_61_Un__iff,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( ( member_set_nat @ C2 @ A2 )
| ( member_set_nat @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_62_Un__iff,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C2 @ A2 )
| ( member_nat @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_63_UnCI,axiom,
! [C2: set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ~ ( member_set_nat @ C2 @ B2 )
=> ( member_set_nat @ C2 @ A2 ) )
=> ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_64_UnCI,axiom,
! [C2: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ A2 ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_65_empty__subsetI,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_66_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_67_subset__empty,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_68_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_69_sup_Obounded__iff,axiom,
! [B: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B @ C2 ) @ A )
= ( ( ord_le6893508408891458716et_nat @ B @ A )
& ( ord_le6893508408891458716et_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_70_sup_Obounded__iff,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
= ( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_71_le__sup__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Z )
& ( ord_le6893508408891458716et_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_72_le__sup__iff,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq_set_nat @ X2 @ Z )
& ( ord_less_eq_set_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_73_insert__subset,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B2 )
= ( ( member_set_nat @ X2 @ B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_74_insert__subset,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( ( member_nat @ X2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_75_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_76_singletonI,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_77_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_78_sup__bot_Oright__neutral,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ bot_bot_set_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_79_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_82_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_83_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_84_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_85_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_86_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_87_Collect__mono__iff,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_88_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_89_set__eq__subset,axiom,
( ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_90_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_91_subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_92_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_93_Collect__mono,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_94_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_95_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_96_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_97_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_98_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_99_equalityD2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_100_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_101_equalityD1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 = B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_102_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_103_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_104_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_105_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_106_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_107_subsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C2 @ A2 )
=> ( member_set_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_108_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_109_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_110_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_111_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_112_ex__in__conv,axiom,
! [A2: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_113_equals0I,axiom,
! [A2: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_114_equals0I,axiom,
! [A2: set_set_nat] :
( ! [Y: set_nat] :
~ ( member_set_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_115_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_116_equals0D,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( A2 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_117_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_118_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_119_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B4: set_nat] :
( ( A2
= ( insert_nat @ A @ B4 ) )
& ~ ( member_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_120_mk__disjoint__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ? [B4: set_set_nat] :
( ( A2
= ( insert_set_nat @ A @ B4 ) )
& ~ ( member_set_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_121_insert__commute,axiom,
! [X2: nat,Y2: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ Y2 @ A2 ) )
= ( insert_nat @ Y2 @ ( insert_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_122_insert__commute,axiom,
! [X2: set_nat,Y2: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ X2 @ ( insert_set_nat @ Y2 @ A2 ) )
= ( insert_set_nat @ Y2 @ ( insert_set_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_123_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_124_insert__eq__iff,axiom,
! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A @ A2 )
=> ( ~ ( member_set_nat @ B @ B2 )
=> ( ( ( insert_set_nat @ A @ A2 )
= ( insert_set_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_set_nat] :
( ( A2
= ( insert_set_nat @ B @ C3 ) )
& ~ ( member_set_nat @ B @ C3 )
& ( B2
= ( insert_set_nat @ A @ C3 ) )
& ~ ( member_set_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_125_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_126_insert__absorb,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_127_insert__ident,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ~ ( member_nat @ X2 @ B2 )
=> ( ( ( insert_nat @ X2 @ A2 )
= ( insert_nat @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_128_insert__ident,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ~ ( member_set_nat @ X2 @ B2 )
=> ( ( ( insert_set_nat @ X2 @ A2 )
= ( insert_set_nat @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_129_Set_Oset__insert,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ! [B4: set_nat] :
( ( A2
= ( insert_nat @ X2 @ B4 ) )
=> ( member_nat @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_130_Set_Oset__insert,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ~ ! [B4: set_set_nat] :
( ( A2
= ( insert_set_nat @ X2 @ B4 ) )
=> ( member_set_nat @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_131_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_132_insertI2,axiom,
! [A: set_nat,B2: set_set_nat,B: set_nat] :
( ( member_set_nat @ A @ B2 )
=> ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_133_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_134_insertI1,axiom,
! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_135_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_136_insertE,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_137_sup__left__commute,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_138_sup__left__commute,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_139_sup_Oleft__commute,axiom,
! [B: set_set_nat,A: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ B @ ( sup_sup_set_set_nat @ A @ C2 ) )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_140_sup_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_141_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] : ( sup_sup_set_set_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_142_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_143_sup_Ocommute,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B5: set_set_nat] : ( sup_sup_set_set_nat @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_144_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_145_sup__assoc,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_146_sup__assoc,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_147_sup_Oassoc,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_148_sup_Oassoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_149_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] : ( sup_sup_set_set_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_150_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_151_inf__sup__aci_I6_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_152_inf__sup__aci_I6_J,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_153_inf__sup__aci_I7_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_154_inf__sup__aci_I7_J,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_155_inf__sup__aci_I8_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_156_inf__sup__aci_I8_J,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_157_Un__left__commute,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) )
= ( sup_sup_set_set_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_158_Un__left__commute,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_159_Un__left__absorb,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_160_Un__left__absorb,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_161_Un__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] : ( sup_sup_set_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_162_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_163_Un__absorb,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_164_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_165_Un__assoc,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) ) ) ).
% Un_assoc
thf(fact_166_Un__assoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% Un_assoc
thf(fact_167_ball__Un,axiom,
! [A2: set_set_nat,B2: set_set_nat,P2: set_nat > $o] :
( ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( P2 @ X3 ) )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_168_ball__Un,axiom,
! [A2: set_nat,B2: set_nat,P2: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P2 @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_169_bex__Un,axiom,
! [A2: set_set_nat,B2: set_set_nat,P2: set_nat > $o] :
( ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P2 @ X3 ) )
| ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_170_bex__Un,axiom,
! [A2: set_nat,B2: set_nat,P2: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P2 @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_171_UnI2,axiom,
! [C2: set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( member_set_nat @ C2 @ B2 )
=> ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_172_UnI2,axiom,
! [C2: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_173_UnI1,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ A2 )
=> ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_174_UnI1,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_175_UnE,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_set_nat @ C2 @ A2 )
=> ( member_set_nat @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_176_UnE,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_177_sup_OcoboundedI2,axiom,
! [C2: set_set_nat,B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ B )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_178_sup_OcoboundedI2,axiom,
! [C2: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_179_sup_OcoboundedI1,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_180_sup_OcoboundedI1,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_181_sup_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B5: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_182_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_183_sup_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B5: set_set_nat,A4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_184_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_185_sup_Ocobounded2,axiom,
! [B: set_set_nat,A: set_set_nat] : ( ord_le6893508408891458716et_nat @ B @ ( sup_sup_set_set_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_186_sup_Ocobounded2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_187_sup_Ocobounded1,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_188_sup_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_189_sup_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B5: set_set_nat,A4: set_set_nat] :
( A4
= ( sup_sup_set_set_nat @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_190_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_191_sup_OboundedI,axiom,
! [B: set_set_nat,A: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ A )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_192_sup_OboundedI,axiom,
! [B: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_193_sup_OboundedE,axiom,
! [B: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ~ ( ord_le6893508408891458716et_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_194_sup_OboundedE,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B @ A )
=> ~ ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_195_sup__absorb2,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( sup_sup_set_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_196_sup__absorb2,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_197_sup__absorb1,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( sup_sup_set_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_198_sup__absorb1,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_199_sup_Oabsorb2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( sup_sup_set_set_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_200_sup_Oabsorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_201_sup_Oabsorb1,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( sup_sup_set_set_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_202_sup_Oabsorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_203_sup__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X2: set_set_nat,Y2: set_set_nat] :
( ! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ ( F @ X @ Y ) )
=> ( ! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y @ ( F @ X @ Y ) )
=> ( ! [X: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( ( ord_le6893508408891458716et_nat @ Z3 @ X )
=> ( ord_le6893508408891458716et_nat @ ( F @ Y @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_204_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y2: set_nat] :
( ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( F @ X @ Y ) )
=> ( ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ Y @ ( F @ X @ Y ) )
=> ( ! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z3 @ X )
=> ( ord_less_eq_set_nat @ ( F @ Y @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_205_sup_OorderI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A
= ( sup_sup_set_set_nat @ A @ B ) )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_206_sup_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_207_sup_OorderE,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A
= ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_208_sup_OorderE,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_209_le__iff__sup,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_210_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_211_sup__least,axiom,
! [Y2: set_set_nat,X2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ Z @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_212_sup__least,axiom,
! [Y2: set_nat,X2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y2 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_213_sup__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ D )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_214_sup__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_215_sup_Omono,axiom,
! [C2: set_set_nat,A: set_set_nat,D: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A )
=> ( ( ord_le6893508408891458716et_nat @ D @ B )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ C2 @ D ) @ ( sup_sup_set_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_216_sup_Omono,axiom,
! [C2: set_nat,A: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_217_le__supI2,axiom,
! [X2: set_set_nat,B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ B )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_218_le__supI2,axiom,
! [X2: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ B )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_219_le__supI1,axiom,
! [X2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ A )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_220_le__supI1,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_221_sup__ge2,axiom,
! [Y2: set_set_nat,X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_222_sup__ge2,axiom,
! [Y2: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_223_sup__ge1,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_224_sup__ge1,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_225_le__supI,axiom,
! [A: set_set_nat,X2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_226_le__supI,axiom,
! [A: set_nat,X2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X2 )
=> ( ( ord_less_eq_set_nat @ B @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_227_le__supE,axiom,
! [A: set_set_nat,B: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ X2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ X2 )
=> ~ ( ord_le6893508408891458716et_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_228_le__supE,axiom,
! [A: set_nat,B: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ X2 )
=> ~ ( ord_less_eq_set_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_229_inf__sup__ord_I3_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_230_inf__sup__ord_I3_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_231_inf__sup__ord_I4_J,axiom,
! [Y2: set_set_nat,X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_232_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_233_subset__insertI2,axiom,
! [A2: set_set_nat,B2: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_234_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_235_subset__insertI,axiom,
! [B2: set_set_nat,A: set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ ( insert_set_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_236_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_237_subset__insert,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ B2 ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_238_subset__insert,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_239_insert__mono,axiom,
! [C: set_set_nat,D2: set_set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ A @ C ) @ ( insert_set_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_240_insert__mono,axiom,
! [C: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_241_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_242_singleton__inject,axiom,
! [A: set_nat,B: set_nat] :
( ( ( insert_set_nat @ A @ bot_bot_set_set_nat )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_243_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_244_insert__not__empty,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ A @ A2 )
!= bot_bot_set_set_nat ) ).
% insert_not_empty
thf(fact_245_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C2 @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_246_doubleton__eq__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat,D: set_nat] :
( ( ( insert_set_nat @ A @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( insert_set_nat @ C2 @ ( insert_set_nat @ D @ bot_bot_set_set_nat ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_247_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_248_singleton__iff,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_249_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_250_singletonD,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_251_subset__Un__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_252_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_253_subset__UnE,axiom,
! [C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
=> ~ ! [A5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ A2 )
=> ! [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ B2 )
=> ( C
!= ( sup_sup_set_set_nat @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_254_subset__UnE,axiom,
! [C: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ~ ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ A2 )
=> ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ B2 )
=> ( C
!= ( sup_sup_set_nat @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_255_Un__absorb2,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_256_Un__absorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_257_Un__absorb1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_258_Un__absorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_259_Un__upper2,axiom,
! [B2: set_set_nat,A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_260_Un__upper2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_261_Un__upper1,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_262_Un__upper1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_263_Un__least,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ C ) ) ) ).
% Un_least
thf(fact_264_Un__least,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C ) ) ) ).
% Un_least
thf(fact_265_Un__mono,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ ( sup_sup_set_set_nat @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_266_Un__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_267_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_268_Un__empty__right,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_269_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_270_Un__empty__left,axiom,
! [B2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_271_subset__singleton__iff,axiom,
! [X4: set_set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( ( X4 = bot_bot_set_set_nat )
| ( X4
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_272_subset__singleton__iff,axiom,
! [X4: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X4 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
| ( X4
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_273_subset__singletonD,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
=> ( ( A2 = bot_bot_set_set_nat )
| ( A2
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_274_subset__singletonD,axiom,
! [A2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_275_singleton__Un__iff,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( ( insert_nat @ X2 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_276_singleton__Un__iff,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ( insert_set_nat @ X2 @ bot_bot_set_set_nat )
= ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_set_nat )
& ( B2
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
| ( ( A2
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
& ( B2 = bot_bot_set_set_nat ) )
| ( ( A2
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
& ( B2
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_277_Un__singleton__iff,axiom,
! [A2: set_nat,B2: set_nat,X2: nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_278_Un__singleton__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat,X2: set_nat] :
( ( ( sup_sup_set_set_nat @ A2 @ B2 )
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= ( ( ( A2 = bot_bot_set_set_nat )
& ( B2
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
| ( ( A2
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
& ( B2 = bot_bot_set_set_nat ) )
| ( ( A2
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
& ( B2
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_279_insert__is__Un,axiom,
( insert_nat
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_280_insert__is__Un,axiom,
( insert_set_nat
= ( ^ [A4: set_nat] : ( sup_sup_set_set_nat @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_281_ball__insert,axiom,
! [A: nat,B2: set_nat,P2: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( insert_nat @ A @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ( P2 @ A )
& ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_insert
thf(fact_282_ball__insert,axiom,
! [A: set_nat,B2: set_set_nat,P2: set_nat > $o] :
( ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( insert_set_nat @ A @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ( P2 @ A )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_insert
thf(fact_283_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_284_the__elem__eq,axiom,
! [X2: set_nat] :
( ( the_elem_set_nat @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_285_dual__order_Orefl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% dual_order.refl
thf(fact_286_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_287_order__refl,axiom,
! [X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_288_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_289_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_290_is__singletonI,axiom,
! [X2: set_nat] : ( is_singleton_set_nat @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ).
% is_singletonI
thf(fact_291_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_292_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ bot_bot_set_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_293_insert__subsetI,axiom,
! [X2: set_nat,A2: set_set_nat,X4: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X4 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_294_insert__subsetI,axiom,
! [X2: nat,A2: set_nat,X4: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_295_subset__emptyI,axiom,
! [A2: set_set_nat] :
( ! [X: set_nat] :
~ ( member_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_296_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_297_bot_Oextremum__uniqueI,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
=> ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_298_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_299_bot_Oextremum__unique,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_300_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_301_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_302_is__singleton__the__elem,axiom,
( is_singleton_set_nat
= ( ^ [A3: set_set_nat] :
( A3
= ( insert_set_nat @ ( the_elem_set_nat @ A3 ) @ bot_bot_set_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_303_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X: nat,Y: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X = Y ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_304_is__singletonI_H,axiom,
! [A2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ! [X: set_nat,Y: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( X = Y ) ) )
=> ( is_singleton_set_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_305_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [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_306_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [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_307_ord__eq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( A = B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_308_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_309_ord__le__eq__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_310_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_311_order__antisym,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_312_order__antisym,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_313_order_Otrans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_314_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_315_order__trans,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ Z )
=> ( ord_le6893508408891458716et_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_316_order__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z )
=> ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_317_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A4 )
& ( ord_le6893508408891458716et_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_318_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_319_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_320_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_321_dual__order_Otrans,axiom,
! [B: set_set_nat,A: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ B )
=> ( ord_le6893508408891458716et_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_322_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_323_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_324_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_325_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
& ( ord_le6893508408891458716et_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_326_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_327_order__subst1,axiom,
! [A: set_set_nat,F: set_set_nat > set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_328_order__subst1,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_329_order__subst1,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_330_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_331_order__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_332_order__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_333_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C2: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_334_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_335_order__eq__refl,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( X2 = Y2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_336_order__eq__refl,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_337_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: set_set_nat > set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_338_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C2: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_339_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C2: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_340_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C2: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_341_ord__le__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_342_ord__le__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_343_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C2: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_344_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_345_order__antisym__conv,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_346_order__antisym__conv,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_347_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_set_nat,K: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( A2
= ( sup_sup_set_set_nat @ K @ A ) )
=> ( ( sup_sup_set_set_nat @ A2 @ B )
= ( sup_sup_set_set_nat @ K @ ( sup_sup_set_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_348_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A2
= ( sup_sup_set_nat @ K @ A ) )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_349_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_set_nat,K: set_set_nat,B: set_set_nat,A: set_set_nat] :
( ( B2
= ( sup_sup_set_set_nat @ K @ B ) )
=> ( ( sup_sup_set_set_nat @ A @ B2 )
= ( sup_sup_set_set_nat @ K @ ( sup_sup_set_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_350_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B2
= ( sup_sup_set_nat @ K @ B ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_351_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X: nat] :
( A2
!= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_352_is__singletonE,axiom,
! [A2: set_set_nat] :
( ( is_singleton_set_nat @ A2 )
=> ~ ! [X: set_nat] :
( A2
!= ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% is_singletonE
thf(fact_353_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X3: nat] :
( A3
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_354_is__singleton__def,axiom,
( is_singleton_set_nat
= ( ^ [A3: set_set_nat] :
? [X3: set_nat] :
( A3
= ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_355_bot_Oextremum,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% bot.extremum
thf(fact_356_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_357_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_358_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_359_Collect__empty__eq__bot,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( P2 = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_360_Collect__empty__eq__bot,axiom,
! [P2: set_nat > $o] :
( ( ( collect_set_nat @ P2 )
= bot_bot_set_set_nat )
= ( P2 = bot_bot_set_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_361_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_362_Set_Ois__empty__def,axiom,
( is_empty_set_nat
= ( ^ [A3: set_set_nat] : ( A3 = bot_bot_set_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_363_sameprod__mono,axiom,
! [X4: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y5 )
=> ( ord_le9131159989063066194et_nat @ ( clique8906516429304539640et_nat @ X4 @ X4 ) @ ( clique8906516429304539640et_nat @ Y5 @ Y5 ) ) ) ).
% sameprod_mono
thf(fact_364_sameprod__mono,axiom,
! [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X4 @ X4 ) @ ( clique6722202388162463298od_nat @ Y5 @ Y5 ) ) ) ).
% sameprod_mono
thf(fact_365_subset__Compl__singleton,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
= ( ~ ( member_set_nat @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_366_subset__Compl__singleton,axiom,
! [A2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_367_GreatestI2__order,axiom,
! [P2: set_set_nat > $o,X2: set_set_nat,Q: set_set_nat > $o] :
( ( P2 @ X2 )
=> ( ! [Y: set_set_nat] :
( ( P2 @ Y )
=> ( ord_le6893508408891458716et_nat @ Y @ X2 ) )
=> ( ! [X: set_set_nat] :
( ( P2 @ X )
=> ( ! [Y6: set_set_nat] :
( ( P2 @ Y6 )
=> ( ord_le6893508408891458716et_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_1279421399067128355et_nat @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_368_GreatestI2__order,axiom,
! [P2: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P2 @ X2 )
=> ( ! [Y: set_nat] :
( ( P2 @ Y )
=> ( ord_less_eq_set_nat @ Y @ X2 ) )
=> ( ! [X: set_nat] :
( ( P2 @ X )
=> ( ! [Y6: set_nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_set_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_369_Greatest__equality,axiom,
! [P2: set_set_nat > $o,X2: set_set_nat] :
( ( P2 @ X2 )
=> ( ! [Y: set_set_nat] :
( ( P2 @ Y )
=> ( ord_le6893508408891458716et_nat @ Y @ X2 ) )
=> ( ( order_1279421399067128355et_nat @ P2 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_370_Greatest__equality,axiom,
! [P2: set_nat > $o,X2: set_nat] :
( ( P2 @ X2 )
=> ( ! [Y: set_nat] :
( ( P2 @ Y )
=> ( ord_less_eq_set_nat @ Y @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P2 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_371_increasingD,axiom,
! [M: set_set_set_nat,F: set_set_nat > set_set_nat,X2: set_set_nat,Y2: set_set_nat] :
( ( measur8782999752332551287et_nat @ M @ F )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( member_set_set_nat @ X2 @ M )
=> ( ( member_set_set_nat @ Y2 @ M )
=> ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_372_increasingD,axiom,
! [M: set_set_set_nat,F: set_set_nat > set_nat,X2: set_set_nat,Y2: set_set_nat] :
( ( measur6219391137901972417et_nat @ M @ F )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( member_set_set_nat @ X2 @ M )
=> ( ( member_set_set_nat @ Y2 @ M )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_373_increasingD,axiom,
! [M: set_set_nat,F: set_nat > set_set_nat,X2: set_nat,Y2: set_nat] :
( ( measur496615480034414785et_nat @ M @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( member_set_nat @ X2 @ M )
=> ( ( member_set_nat @ Y2 @ M )
=> ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_374_increasingD,axiom,
! [M: set_set_nat,F: set_nat > set_nat,X2: set_nat,Y2: set_nat] :
( ( measur5248428813077667851et_nat @ M @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( member_set_nat @ X2 @ M )
=> ( ( member_set_nat @ Y2 @ M )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_375_ComplI,axiom,
! [C2: nat,A2: set_nat] :
( ~ ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_376_ComplI,axiom,
! [C2: set_nat,A2: set_set_nat] :
( ~ ( member_set_nat @ C2 @ A2 )
=> ( member_set_nat @ C2 @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_377_Compl__iff,axiom,
! [C2: nat,A2: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( ~ ( member_nat @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_378_Compl__iff,axiom,
! [C2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ C2 @ ( uminus613421341184616069et_nat @ A2 ) )
= ( ~ ( member_set_nat @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_379_compl__le__compl__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ X2 ) @ ( uminus613421341184616069et_nat @ Y2 ) )
= ( ord_le6893508408891458716et_nat @ Y2 @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_380_compl__le__compl__iff,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( uminus5710092332889474511et_nat @ Y2 ) )
= ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_381_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_382_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_383_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_384_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_385_ComplD,axiom,
! [C2: nat,A2: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) )
=> ~ ( member_nat @ C2 @ A2 ) ) ).
% ComplD
thf(fact_386_ComplD,axiom,
! [C2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ C2 @ ( uminus613421341184616069et_nat @ A2 ) )
=> ~ ( member_set_nat @ C2 @ A2 ) ) ).
% ComplD
thf(fact_387_compl__le__swap2,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ Y2 ) @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ X2 ) @ Y2 ) ) ).
% compl_le_swap2
thf(fact_388_compl__le__swap2,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y2 ) @ X2 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y2 ) ) ).
% compl_le_swap2
thf(fact_389_compl__le__swap1,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ ( uminus613421341184616069et_nat @ X2 ) )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( uminus613421341184616069et_nat @ Y2 ) ) ) ).
% compl_le_swap1
thf(fact_390_compl__le__swap1,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ ( uminus5710092332889474511et_nat @ X2 ) )
=> ( ord_less_eq_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y2 ) ) ) ).
% compl_le_swap1
thf(fact_391_compl__mono,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ Y2 ) @ ( uminus613421341184616069et_nat @ X2 ) ) ) ).
% compl_mono
thf(fact_392_compl__mono,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y2 ) @ ( uminus5710092332889474511et_nat @ X2 ) ) ) ).
% compl_mono
thf(fact_393_subset__Compl__self__eq,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_394_subset__Compl__self__eq,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_395_increasing__def,axiom,
( measur8782999752332551287et_nat
= ( ^ [M2: set_set_set_nat,Mu: set_set_nat > set_set_nat] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ M2 )
=> ! [Y4: set_set_nat] :
( ( member_set_set_nat @ Y4 @ M2 )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( Mu @ X3 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_396_increasing__def,axiom,
( measur6219391137901972417et_nat
= ( ^ [M2: set_set_set_nat,Mu: set_set_nat > set_nat] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ M2 )
=> ! [Y4: set_set_nat] :
( ( member_set_set_nat @ Y4 @ M2 )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( Mu @ X3 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_397_increasing__def,axiom,
( measur496615480034414785et_nat
= ( ^ [M2: set_set_nat,Mu: set_nat > set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ M2 )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ M2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( Mu @ X3 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_398_increasing__def,axiom,
( measur5248428813077667851et_nat
= ( ^ [M2: set_set_nat,Mu: set_nat > set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ M2 )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ M2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( Mu @ X3 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_399_verit__comp__simplify1_I2_J,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_400_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_401_Compl__insert,axiom,
! [X2: nat,A2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X2 @ A2 ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_402_Compl__insert,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( uminus613421341184616069et_nat @ ( insert_set_nat @ X2 @ A2 ) )
= ( minus_2163939370556025621et_nat @ ( uminus613421341184616069et_nat @ A2 ) @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ).
% Compl_insert
thf(fact_403_disjoint__eq__subset__Compl,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_404_disjoint__eq__subset__Compl,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_405_sup__bot_Osemilattice__neutr__axioms,axiom,
semila1241773964035338532et_nat @ sup_sup_set_nat @ bot_bot_set_nat ).
% sup_bot.semilattice_neutr_axioms
thf(fact_406_sup__bot_Osemilattice__neutr__axioms,axiom,
semila7398651959140203994et_nat @ sup_sup_set_set_nat @ bot_bot_set_set_nat ).
% sup_bot.semilattice_neutr_axioms
thf(fact_407_Un__Pow__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ ( pow_set_nat @ A2 ) @ ( pow_set_nat @ B2 ) ) @ ( pow_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% Un_Pow_subset
thf(fact_408_Un__Pow__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B2 ) ) @ ( pow_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_Pow_subset
thf(fact_409_le__rel__bool__arg__iff,axiom,
( ord_le6539261115178940645et_nat
= ( ^ [X5: $o > set_set_nat,Y7: $o > set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le6893508408891458716et_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_410_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X5: $o > set_nat,Y7: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_set_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_411_le__left__mono,axiom,
! [X2: set_set_nat,Y2: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ A )
=> ( ord_le6893508408891458716et_nat @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_412_le__left__mono,axiom,
! [X2: set_nat,Y2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_413_IntI,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_414_IntI,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ A2 )
=> ( ( member_set_nat @ C2 @ B2 )
=> ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_415_Int__iff,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C2 @ A2 )
& ( member_nat @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_416_Int__iff,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( ( member_set_nat @ C2 @ A2 )
& ( member_set_nat @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_417_DiffI,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ~ ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_418_DiffI,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ A2 )
=> ( ~ ( member_set_nat @ C2 @ B2 )
=> ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_419_Diff__iff,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C2 @ A2 )
& ~ ( member_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_420_Diff__iff,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( ( member_set_nat @ C2 @ A2 )
& ~ ( member_set_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_421_inf_Obounded__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) )
= ( ( ord_le6893508408891458716et_nat @ A @ B )
& ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_422_inf_Obounded__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_423_le__inf__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z ) )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
& ( ord_le6893508408891458716et_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_424_le__inf__iff,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z ) )
= ( ( ord_less_eq_set_nat @ X2 @ Y2 )
& ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_425_inf__bot__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_426_inf__bot__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% inf_bot_right
thf(fact_427_inf__bot__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_428_inf__bot__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X2 )
= bot_bot_set_set_nat ) ).
% inf_bot_left
thf(fact_429_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_430_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_431_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_432_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X2 )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_433_Int__subset__iff,axiom,
! [C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( ( ord_le6893508408891458716et_nat @ C @ A2 )
& ( ord_le6893508408891458716et_nat @ C @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_434_Int__subset__iff,axiom,
! [C: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_set_nat @ C @ A2 )
& ( ord_less_eq_set_nat @ C @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_435_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_436_Int__insert__right__if1,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_437_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_438_Int__insert__right__if0,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_439_insert__inter__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_440_insert__inter__insert,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ ( insert_set_nat @ A @ B2 ) )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_441_Int__insert__left__if1,axiom,
! [A: nat,C: set_nat,B2: set_nat] :
( ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_442_Int__insert__left__if1,axiom,
! [A: set_nat,C: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A @ C )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_443_Int__insert__left__if0,axiom,
! [A: nat,C: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C )
= ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_444_Int__insert__left__if0,axiom,
! [A: set_nat,C: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A @ C )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C )
= ( inf_inf_set_set_nat @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_445_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_446_Diff__cancel,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ A2 )
= bot_bot_set_set_nat ) ).
% Diff_cancel
thf(fact_447_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_448_empty__Diff,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ bot_bot_set_set_nat @ A2 )
= bot_bot_set_set_nat ) ).
% empty_Diff
thf(fact_449_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_450_Diff__empty,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_451_sup__inf__absorb,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_452_sup__inf__absorb,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_453_inf__sup__absorb,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_454_inf__sup__absorb,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_455_insert__Diff1,axiom,
! [X2: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_456_insert__Diff1,axiom,
! [X2: set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ B2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B2 )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_457_Diff__insert0,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_458_Diff__insert0,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ B2 ) )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_459_Un__Int__eq_I1_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_460_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_461_Un__Int__eq_I2_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_462_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_463_Un__Int__eq_I3_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ S @ ( sup_sup_set_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_464_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_465_Un__Int__eq_I4_J,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( inf_inf_set_set_nat @ T2 @ ( sup_sup_set_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_466_Un__Int__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_467_Int__Un__eq_I1_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_468_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_469_Int__Un__eq_I2_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_470_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_471_Int__Un__eq_I3_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ S @ ( inf_inf_set_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_472_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_473_Int__Un__eq_I4_J,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( sup_sup_set_set_nat @ T2 @ ( inf_inf_set_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_474_Int__Un__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_475_Un__Diff__cancel2,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_set_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_476_Un__Diff__cancel2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_477_Un__Diff__cancel,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) )
= ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_478_Un__Diff__cancel,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_479_Pow__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( member_set_set_nat @ A2 @ ( pow_set_nat @ B2 ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_480_Pow__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( member_set_nat @ A2 @ ( pow_nat @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_481_PowI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( member_set_set_nat @ A2 @ ( pow_set_nat @ B2 ) ) ) ).
% PowI
thf(fact_482_PowI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( member_set_nat @ A2 @ ( pow_nat @ B2 ) ) ) ).
% PowI
thf(fact_483_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_484_Pow__empty,axiom,
( ( pow_set_nat @ bot_bot_set_set_nat )
= ( insert_set_set_nat @ bot_bot_set_set_nat @ bot_bo7198184520161983622et_nat ) ) ).
% Pow_empty
thf(fact_485_Pow__singleton__iff,axiom,
! [X4: set_set_nat,Y5: set_set_nat] :
( ( ( pow_set_nat @ X4 )
= ( insert_set_set_nat @ Y5 @ bot_bo7198184520161983622et_nat ) )
= ( ( X4 = bot_bot_set_set_nat )
& ( Y5 = bot_bot_set_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_486_Pow__singleton__iff,axiom,
! [X4: set_nat,Y5: set_nat] :
( ( ( pow_nat @ X4 )
= ( insert_set_nat @ Y5 @ bot_bot_set_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y5 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_487_inf__compl__bot__left1,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( inf_inf_set_nat @ X2 @ Y2 ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left1
thf(fact_488_inf__compl__bot__left1,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( uminus613421341184616069et_nat @ X2 ) @ ( inf_inf_set_set_nat @ X2 @ Y2 ) )
= bot_bot_set_set_nat ) ).
% inf_compl_bot_left1
thf(fact_489_inf__compl__bot__left2,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y2 ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left2
thf(fact_490_inf__compl__bot__left2,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ ( uminus613421341184616069et_nat @ X2 ) @ Y2 ) )
= bot_bot_set_set_nat ) ).
% inf_compl_bot_left2
thf(fact_491_inf__compl__bot__right,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ ( uminus5710092332889474511et_nat @ X2 ) ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_right
thf(fact_492_inf__compl__bot__right,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ ( uminus613421341184616069et_nat @ X2 ) ) )
= bot_bot_set_set_nat ) ).
% inf_compl_bot_right
thf(fact_493_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ X2 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_left
thf(fact_494_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( uminus613421341184616069et_nat @ X2 ) @ X2 )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_cancel_left
thf(fact_495_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ X2 ) )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_right
thf(fact_496_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( uminus613421341184616069et_nat @ X2 ) )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_cancel_right
thf(fact_497_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_498_insert__disjoint_I1_J,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ B2 )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A @ B2 )
& ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_499_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member_nat @ A @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_500_insert__disjoint_I2_J,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member_set_nat @ A @ B2 )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_501_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_502_disjoint__insert_I1_J,axiom,
! [B2: set_set_nat,A: set_nat,A2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ B2 @ ( insert_set_nat @ A @ A2 ) )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A @ B2 )
& ( ( inf_inf_set_set_nat @ B2 @ A2 )
= bot_bot_set_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_503_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) )
= ( ~ ( member_nat @ B @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_504_disjoint__insert_I2_J,axiom,
! [A2: set_set_nat,B: set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) )
= ( ~ ( member_set_nat @ B @ A2 )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_505_Diff__eq__empty__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_506_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_507_boolean__algebra_Ode__Morgan__conj,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( uminus613421341184616069et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ X2 ) @ ( uminus613421341184616069et_nat @ Y2 ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_508_boolean__algebra_Ode__Morgan__conj,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( uminus5710092332889474511et_nat @ Y2 ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_509_boolean__algebra_Ode__Morgan__disj,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( uminus613421341184616069et_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= ( inf_inf_set_set_nat @ ( uminus613421341184616069et_nat @ X2 ) @ ( uminus613421341184616069et_nat @ Y2 ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_510_boolean__algebra_Ode__Morgan__disj,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) )
= ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( uminus5710092332889474511et_nat @ Y2 ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_511_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_512_insert__Diff__single,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( insert_set_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_513_Diff__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_514_Diff__disjoint,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) )
= bot_bot_set_set_nat ) ).
% Diff_disjoint
thf(fact_515_Compl__disjoint2,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ A2 )
= bot_bot_set_nat ) ).
% Compl_disjoint2
thf(fact_516_Compl__disjoint2,axiom,
! [A2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( uminus613421341184616069et_nat @ A2 ) @ A2 )
= bot_bot_set_set_nat ) ).
% Compl_disjoint2
thf(fact_517_Compl__disjoint,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= bot_bot_set_nat ) ).
% Compl_disjoint
thf(fact_518_Compl__disjoint,axiom,
! [A2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ ( uminus613421341184616069et_nat @ A2 ) )
= bot_bot_set_set_nat ) ).
% Compl_disjoint
thf(fact_519_Compl__Diff__eq,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( uminus613421341184616069et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ A2 ) @ B2 ) ) ).
% Compl_Diff_eq
thf(fact_520_Compl__Diff__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ B2 ) ) ).
% Compl_Diff_eq
thf(fact_521_Pow__not__empty,axiom,
! [A2: set_nat] :
( ( pow_nat @ A2 )
!= bot_bot_set_set_nat ) ).
% Pow_not_empty
thf(fact_522_Int__Diff__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_523_Int__Diff__disjoint,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= bot_bot_set_set_nat ) ).
% Int_Diff_disjoint
thf(fact_524_Diff__triv,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_525_Diff__triv,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
=> ( ( minus_2163939370556025621et_nat @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_526_Un__Diff__Int,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_527_Un__Diff__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_528_Int__Diff__Un,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_529_Int__Diff__Un,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_530_Diff__Int,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ ( minus_2163939370556025621et_nat @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_531_Diff__Int,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_532_Diff__Un,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) )
= ( inf_inf_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ ( minus_2163939370556025621et_nat @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_533_Diff__Un,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_534_IntE,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ~ ( member_nat @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_535_IntE,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_set_nat @ C2 @ A2 )
=> ~ ( member_set_nat @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_536_DiffE,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_537_DiffE,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ~ ( ( member_set_nat @ C2 @ A2 )
=> ( member_set_nat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_538_IntD1,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% IntD1
thf(fact_539_IntD1,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ( member_set_nat @ C2 @ A2 ) ) ).
% IntD1
thf(fact_540_IntD2,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C2 @ B2 ) ) ).
% IntD2
thf(fact_541_IntD2,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ( member_set_nat @ C2 @ B2 ) ) ).
% IntD2
thf(fact_542_DiffD1,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_543_DiffD1,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ( member_set_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_544_DiffD2,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_545_DiffD2,axiom,
! [C2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
=> ~ ( member_set_nat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_546_Pow__top,axiom,
! [A2: set_nat] : ( member_set_nat @ A2 @ ( pow_nat @ A2 ) ) ).
% Pow_top
thf(fact_547_PowD,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( member_set_set_nat @ A2 @ ( pow_set_nat @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% PowD
thf(fact_548_PowD,axiom,
! [A2: set_nat,B2: set_nat] :
( ( member_set_nat @ A2 @ ( pow_nat @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% PowD
thf(fact_549_Pow__bottom,axiom,
! [B2: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B2 ) ) ).
% Pow_bottom
thf(fact_550_Pow__bottom,axiom,
! [B2: set_set_nat] : ( member_set_set_nat @ bot_bot_set_set_nat @ ( pow_set_nat @ B2 ) ) ).
% Pow_bottom
thf(fact_551_inf_OcoboundedI2,axiom,
! [B: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_552_inf_OcoboundedI2,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_553_inf_OcoboundedI1,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_554_inf_OcoboundedI1,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_555_inf_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B5: set_set_nat,A4: set_set_nat] :
( ( inf_inf_set_set_nat @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_556_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_557_inf_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B5: set_set_nat] :
( ( inf_inf_set_set_nat @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_558_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_559_inf_Ocobounded2,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_560_inf_Ocobounded2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_561_inf_Ocobounded1,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_562_inf_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_563_inf_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B5: set_set_nat] :
( A4
= ( inf_inf_set_set_nat @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_564_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( A4
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_565_inf__greatest,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Z )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_566_inf__greatest,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_567_inf_OboundedI,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_568_inf_OboundedI,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_569_inf_OboundedE,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ~ ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_570_inf_OboundedE,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_571_inf__absorb2,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( inf_inf_set_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_572_inf__absorb2,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_573_inf__absorb1,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( inf_inf_set_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_574_inf__absorb1,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_575_inf_Oabsorb2,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( inf_inf_set_set_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_576_inf_Oabsorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_577_inf_Oabsorb1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( inf_inf_set_set_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_578_inf_Oabsorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_579_le__iff__inf,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y4: set_set_nat] :
( ( inf_inf_set_set_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_580_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_581_inf__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X2: set_set_nat,Y2: set_set_nat] :
( ! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F @ X @ Y ) @ X )
=> ( ! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F @ X @ Y ) @ Y )
=> ( ! [X: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ X @ Z3 )
=> ( ord_le6893508408891458716et_nat @ X @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_582_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y2: set_nat] :
( ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F @ X @ Y ) @ X )
=> ( ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F @ X @ Y ) @ Y )
=> ( ! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z3 )
=> ( ord_less_eq_set_nat @ X @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_583_inf_OorderI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A
= ( inf_inf_set_set_nat @ A @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_584_inf_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( inf_inf_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_585_inf_OorderE,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( A
= ( inf_inf_set_set_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_586_inf_OorderE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( A
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_587_le__infI2,axiom,
! [B: set_set_nat,X2: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_588_le__infI2,axiom,
! [B: set_nat,X2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_589_le__infI1,axiom,
! [A: set_set_nat,X2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_590_le__infI1,axiom,
! [A: set_nat,X2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_591_inf__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ D )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_592_inf__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_593_le__infI,axiom,
! [X2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ A )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ B )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_594_le__infI,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ B )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_595_le__infE,axiom,
! [X2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ A @ B ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ X2 @ A )
=> ~ ( ord_le6893508408891458716et_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_596_le__infE,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_set_nat @ X2 @ A )
=> ~ ( ord_less_eq_set_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_597_inf__le2,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_598_inf__le2,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_599_inf__le1,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_600_inf__le1,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_601_inf__sup__ord_I1_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_602_inf__sup__ord_I1_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_603_inf__sup__ord_I2_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_604_inf__sup__ord_I2_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_605_sup__inf__distrib2,axiom,
! [Y2: set_set_nat,Z: set_set_nat,X2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y2 @ Z ) @ X2 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y2 @ X2 ) @ ( sup_sup_set_set_nat @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_606_sup__inf__distrib2,axiom,
! [Y2: set_nat,Z: set_nat,X2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ Z ) @ X2 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ X2 ) @ ( sup_sup_set_nat @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_607_sup__inf__distrib1,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_608_sup__inf__distrib1,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_609_inf__sup__distrib2,axiom,
! [Y2: set_set_nat,Z: set_set_nat,X2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y2 @ Z ) @ X2 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y2 @ X2 ) @ ( inf_inf_set_set_nat @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_610_inf__sup__distrib2,axiom,
! [Y2: set_nat,Z: set_nat,X2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ Z ) @ X2 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ X2 ) @ ( inf_inf_set_nat @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_611_inf__sup__distrib1,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_612_inf__sup__distrib1,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_613_distrib__imp2,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ! [X: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z3 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y ) @ ( sup_sup_set_set_nat @ X @ Z3 ) ) )
=> ( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_614_distrib__imp2,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z3 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z3 ) ) )
=> ( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_615_distrib__imp1,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ! [X: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y @ Z3 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ ( inf_inf_set_set_nat @ X @ Z3 ) ) )
=> ( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_616_distrib__imp1,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z3 ) ) )
=> ( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_617_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_set_nat,Z: set_set_nat,X2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y2 @ Z ) @ X2 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y2 @ X2 ) @ ( sup_sup_set_set_nat @ Z @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_618_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_nat,Z: set_nat,X2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ Z ) @ X2 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ X2 ) @ ( sup_sup_set_nat @ Z @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_619_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_set_nat,Z: set_set_nat,X2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y2 @ Z ) @ X2 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y2 @ X2 ) @ ( inf_inf_set_set_nat @ Z @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_620_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_nat,Z: set_nat,X2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ Z ) @ X2 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ X2 ) @ ( inf_inf_set_nat @ Z @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_621_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_622_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_623_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_624_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_625_Int__Collect__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,P2: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( P2 @ X )
=> ( Q @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P2 ) ) @ ( inf_inf_set_set_nat @ B2 @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_626_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( P2 @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_627_Int__greatest,axiom,
! [C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ B2 )
=> ( ord_le6893508408891458716et_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_628_Int__greatest,axiom,
! [C: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_629_Int__absorb2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_630_Int__absorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_631_Int__absorb1,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_632_Int__absorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_633_Int__lower2,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_634_Int__lower2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_635_Int__lower1,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_636_Int__lower1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_637_Int__mono,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_638_Int__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_639_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_640_disjoint__iff__not__equal,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_641_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_642_Int__empty__right,axiom,
! [A2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% Int_empty_right
thf(fact_643_Int__empty__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_644_Int__empty__left,axiom,
! [B2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ B2 )
= bot_bot_set_set_nat ) ).
% Int_empty_left
thf(fact_645_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_646_disjoint__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ~ ( member_set_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_647_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_nat @ X @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_648_Int__emptyI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ~ ( member_set_nat @ X @ B2 ) )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_649_double__diff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ( minus_2163939370556025621et_nat @ B2 @ ( minus_2163939370556025621et_nat @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_650_double__diff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_651_Diff__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_652_Diff__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_653_Diff__mono,axiom,
! [A2: set_set_nat,C: set_set_nat,D2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ ( minus_2163939370556025621et_nat @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_654_Diff__mono,axiom,
! [A2: set_nat,C: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_655_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_656_Int__insert__right,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_657_Int__insert__left,axiom,
! [A: nat,C: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C ) ) ) )
& ( ~ ( member_nat @ A @ C )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C )
= ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_658_Int__insert__left,axiom,
! [A: set_nat,C: set_set_nat,B2: set_set_nat] :
( ( ( member_set_nat @ A @ C )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ B2 @ C ) ) ) )
& ( ~ ( member_set_nat @ A @ C )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C )
= ( inf_inf_set_set_nat @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_659_insert__Diff__if,axiom,
! [X2: nat,B2: set_nat,A2: set_nat] :
( ( ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( insert_nat @ X2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_660_insert__Diff__if,axiom,
! [X2: set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ( member_set_nat @ X2 @ B2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B2 )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
& ( ~ ( member_set_nat @ X2 @ B2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B2 )
= ( insert_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_661_Un__Int__distrib2,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B2 @ C ) @ A2 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B2 @ A2 ) @ ( sup_sup_set_set_nat @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_662_Un__Int__distrib2,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ C ) @ A2 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ A2 ) @ ( sup_sup_set_nat @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_663_Int__Un__distrib2,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B2 @ C ) @ A2 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B2 @ A2 ) @ ( inf_inf_set_set_nat @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_664_Int__Un__distrib2,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ A2 ) @ ( inf_inf_set_nat @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_665_Un__Int__distrib,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ ( sup_sup_set_set_nat @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_666_Un__Int__distrib,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_667_Int__Un__distrib,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_668_Int__Un__distrib,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_669_Un__Int__crazy,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ B2 @ C ) ) @ ( inf_inf_set_set_nat @ C @ A2 ) )
= ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ ( sup_sup_set_set_nat @ B2 @ C ) ) @ ( sup_sup_set_set_nat @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_670_Un__Int__crazy,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ B2 @ C ) ) @ ( inf_inf_set_nat @ C @ A2 ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ B2 @ C ) ) @ ( sup_sup_set_nat @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_671_Un__Diff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( minus_2163939370556025621et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ C ) @ ( minus_2163939370556025621et_nat @ B2 @ C ) ) ) ).
% Un_Diff
thf(fact_672_Un__Diff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C ) @ ( minus_minus_set_nat @ B2 @ C ) ) ) ).
% Un_Diff
thf(fact_673_Pow__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( pow_set_nat @ A2 ) @ ( pow_set_nat @ B2 ) ) ) ).
% Pow_mono
thf(fact_674_Pow__mono,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B2 ) ) ) ).
% Pow_mono
thf(fact_675_inf__cancel__left1,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X2 @ A ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left1
thf(fact_676_inf__cancel__left1,axiom,
! [X2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ A ) @ ( inf_inf_set_set_nat @ ( uminus613421341184616069et_nat @ X2 ) @ B ) )
= bot_bot_set_set_nat ) ).
% inf_cancel_left1
thf(fact_677_inf__cancel__left2,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ A ) @ ( inf_inf_set_nat @ X2 @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left2
thf(fact_678_inf__cancel__left2,axiom,
! [X2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ ( uminus613421341184616069et_nat @ X2 ) @ A ) @ ( inf_inf_set_set_nat @ X2 @ B ) )
= bot_bot_set_set_nat ) ).
% inf_cancel_left2
thf(fact_679_distrib__sup__le,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z ) ) @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_680_distrib__sup__le,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_681_distrib__inf__le,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) @ ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_682_distrib__inf__le,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) @ ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_683_diff__shunt__var,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ X2 @ Y2 )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ X2 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_684_diff__shunt__var,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_685_subset__Diff__insert,axiom,
! [A2: set_set_nat,B2: set_set_nat,X2: set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X2 @ C ) ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C ) )
& ~ ( member_set_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_686_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X2: nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X2 @ C ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C ) )
& ~ ( member_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_687_Diff__insert__absorb,axiom,
! [X2: nat,A2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_688_Diff__insert__absorb,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_689_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_690_Diff__insert2,axiom,
! [A2: set_set_nat,A: set_nat,B2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_691_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_692_insert__Diff,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_693_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_694_Diff__insert,axiom,
! [A2: set_set_nat,A: set_nat,B2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ).
% Diff_insert
thf(fact_695_Un__Int__assoc__eq,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C )
= ( inf_inf_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) ) )
= ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_696_Un__Int__assoc__eq,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C )
= ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) )
= ( ord_less_eq_set_nat @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_697_Diff__subset__conv,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ C )
= ( ord_le6893508408891458716et_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) ) ) ).
% Diff_subset_conv
thf(fact_698_Diff__subset__conv,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% Diff_subset_conv
thf(fact_699_Diff__partition,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( sup_sup_set_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_700_Diff__partition,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_701_Compl__Int,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( uminus613421341184616069et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ A2 ) @ ( uminus613421341184616069et_nat @ B2 ) ) ) ).
% Compl_Int
thf(fact_702_Compl__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% Compl_Int
thf(fact_703_Compl__Un,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( uminus613421341184616069et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_set_nat @ ( uminus613421341184616069et_nat @ A2 ) @ ( uminus613421341184616069et_nat @ B2 ) ) ) ).
% Compl_Un
thf(fact_704_Compl__Un,axiom,
! [A2: set_nat,B2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% Compl_Un
thf(fact_705_inf__shunt,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ X2 @ Y2 )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ X2 @ ( uminus613421341184616069et_nat @ Y2 ) ) ) ).
% inf_shunt
thf(fact_706_inf__shunt,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ( inf_inf_set_nat @ X2 @ Y2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y2 ) ) ) ).
% inf_shunt
thf(fact_707_sup__neg__inf,axiom,
! [P: set_set_nat,Q2: set_set_nat,R: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ P @ ( sup_sup_set_set_nat @ Q2 @ R ) )
= ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ P @ ( uminus613421341184616069et_nat @ Q2 ) ) @ R ) ) ).
% sup_neg_inf
thf(fact_708_sup__neg__inf,axiom,
! [P: set_nat,Q2: set_nat,R: set_nat] :
( ( ord_less_eq_set_nat @ P @ ( sup_sup_set_nat @ Q2 @ R ) )
= ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P @ ( uminus5710092332889474511et_nat @ Q2 ) ) @ R ) ) ).
% sup_neg_inf
thf(fact_709_shunt2,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ ( uminus613421341184616069et_nat @ Y2 ) ) @ Z )
= ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% shunt2
thf(fact_710_shunt2,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y2 ) ) @ Z )
= ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% shunt2
thf(fact_711_shunt1,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ Z )
= ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ Y2 ) @ Z ) ) ) ).
% shunt1
thf(fact_712_shunt1,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ Z )
= ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y2 ) @ Z ) ) ) ).
% shunt1
thf(fact_713_Diff__single__insert,axiom,
! [A2: set_set_nat,X2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_714_Diff__single__insert,axiom,
! [A2: set_nat,X2: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_715_subset__insert__iff,axiom,
! [A2: set_set_nat,X2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ B2 ) )
= ( ( ( member_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 ) )
& ( ~ ( member_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_716_subset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_717_self__bounded__weaken__right,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B @ A ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% self_bounded_weaken_right
thf(fact_718_self__bounded__weaken__right,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ A ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% self_bounded_weaken_right
thf(fact_719_self__bounded__weaken__left,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ A @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% self_bounded_weaken_left
thf(fact_720_self__bounded__weaken__left,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% self_bounded_weaken_left
thf(fact_721_remove__def,axiom,
( remove_nat
= ( ^ [X3: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_722_remove__def,axiom,
( remove_set_nat
= ( ^ [X3: set_nat,A3: set_set_nat] : ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ) ).
% remove_def
thf(fact_723_psubset__insert__iff,axiom,
! [A2: set_set_nat,X2: set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X2 @ B2 ) )
= ( ( ( member_set_nat @ X2 @ B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) )
& ( ~ ( member_set_nat @ X2 @ B2 )
=> ( ( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 ) )
& ( ~ ( member_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_724_psubset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( ( ( member_nat @ X2 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat @ X2 @ B2 )
=> ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_725_algebra__iff__Un,axiom,
( sigma_5697435980195335136et_nat
= ( ^ [Omega: set_set_nat,M2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ M2 @ ( pow_set_nat @ Omega ) )
& ( member_set_set_nat @ bot_bot_set_set_nat @ M2 )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ M2 )
=> ( member_set_set_nat @ ( minus_2163939370556025621et_nat @ Omega @ X3 ) @ M2 ) )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ M2 )
=> ! [Y4: set_set_nat] :
( ( member_set_set_nat @ Y4 @ M2 )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ X3 @ Y4 ) @ M2 ) ) ) ) ) ) ).
% algebra_iff_Un
thf(fact_726_algebra__iff__Un,axiom,
( sigma_algebra_nat
= ( ^ [Omega: set_nat,M2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ M2 @ ( pow_nat @ Omega ) )
& ( member_set_nat @ bot_bot_set_nat @ M2 )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ M2 )
=> ( member_set_nat @ ( minus_minus_set_nat @ Omega @ X3 ) @ M2 ) )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ M2 )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ M2 )
=> ( member_set_nat @ ( sup_sup_set_nat @ X3 @ Y4 ) @ M2 ) ) ) ) ) ) ).
% algebra_iff_Un
thf(fact_727_algebra__iff__Int,axiom,
( sigma_5697435980195335136et_nat
= ( ^ [Omega: set_set_nat,M2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ M2 @ ( pow_set_nat @ Omega ) )
& ( member_set_set_nat @ bot_bot_set_set_nat @ M2 )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ M2 )
=> ( member_set_set_nat @ ( minus_2163939370556025621et_nat @ Omega @ X3 ) @ M2 ) )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ M2 )
=> ! [Y4: set_set_nat] :
( ( member_set_set_nat @ Y4 @ M2 )
=> ( member_set_set_nat @ ( inf_inf_set_set_nat @ X3 @ Y4 ) @ M2 ) ) ) ) ) ) ).
% algebra_iff_Int
thf(fact_728_algebra__iff__Int,axiom,
( sigma_algebra_nat
= ( ^ [Omega: set_nat,M2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ M2 @ ( pow_nat @ Omega ) )
& ( member_set_nat @ bot_bot_set_nat @ M2 )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ M2 )
=> ( member_set_nat @ ( minus_minus_set_nat @ Omega @ X3 ) @ M2 ) )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ M2 )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ M2 )
=> ( member_set_nat @ ( inf_inf_set_nat @ X3 @ Y4 ) @ M2 ) ) ) ) ) ) ).
% algebra_iff_Int
thf(fact_729_member__remove,axiom,
! [X2: nat,Y2: nat,A2: set_nat] :
( ( member_nat @ X2 @ ( remove_nat @ Y2 @ A2 ) )
= ( ( member_nat @ X2 @ A2 )
& ( X2 != Y2 ) ) ) ).
% member_remove
thf(fact_730_member__remove,axiom,
! [X2: set_nat,Y2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ ( remove_set_nat @ Y2 @ A2 ) )
= ( ( member_set_nat @ X2 @ A2 )
& ( X2 != Y2 ) ) ) ).
% member_remove
thf(fact_731_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_732_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_733_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_734_psubsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C2 @ A2 )
=> ( member_set_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_735_leD,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ~ ( ord_less_set_set_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_736_leD,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_737_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_738_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_739_antisym__conv1,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ~ ( ord_less_set_set_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_740_antisym__conv1,axiom,
! [X2: set_nat,Y2: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_741_antisym__conv2,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_set_set_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_742_antisym__conv2,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_743_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_744_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_745_order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B5: set_set_nat] :
( ( ord_less_set_set_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_746_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_747_order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [A4: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_748_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_749_order_Ostrict__trans1,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ B @ C2 )
=> ( ord_less_set_set_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_750_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_751_order_Ostrict__trans2,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_less_set_set_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_752_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_753_order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [A4: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
& ~ ( ord_le6893508408891458716et_nat @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_754_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_755_dual__order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B5: set_set_nat,A4: set_set_nat] :
( ( ord_less_set_set_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_756_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_757_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [B5: set_set_nat,A4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_758_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_759_dual__order_Ostrict__trans1,axiom,
! [B: set_set_nat,A: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_less_set_set_nat @ C2 @ B )
=> ( ord_less_set_set_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_760_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C2 @ B )
=> ( ord_less_set_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_761_dual__order_Ostrict__trans2,axiom,
! [B: set_set_nat,A: set_set_nat,C2: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ B )
=> ( ord_less_set_set_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_762_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_set_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_763_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [B5: set_set_nat,A4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A4 )
& ~ ( ord_le6893508408891458716et_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_764_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_765_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_766_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_767_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_768_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_769_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_770_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_771_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_772_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_773_order__less__imp__le,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_774_order__less__imp__le,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_775_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_776_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_777_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_778_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_779_order__le__less__trans,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_less_set_set_nat @ Y2 @ Z )
=> ( ord_less_set_set_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_780_order__le__less__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_781_order__less__le__trans,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ Z )
=> ( ord_less_set_set_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_782_order__less__le__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_783_order__le__less__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_784_order__le__less__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_785_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C2: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_786_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_787_order__less__le__subst1,axiom,
! [A: set_set_nat,F: set_set_nat > set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_less_set_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_788_order__less__le__subst1,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_789_order__less__le__subst1,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_790_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_791_order__le__imp__less__or__eq,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_less_set_set_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_792_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_set_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_793_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_794_bot_Oextremum__strict,axiom,
! [A: set_set_nat] :
~ ( ord_less_set_set_nat @ A @ bot_bot_set_set_nat ) ).
% bot.extremum_strict
thf(fact_795_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_796_bot_Onot__eq__extremum,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
= ( ord_less_set_set_nat @ bot_bot_set_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_797_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_798_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_799_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_800_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_801_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_802_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_803_psubset__subset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_less_set_set_nat @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_804_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_805_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ~ ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_806_subset__not__subset__eq,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 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_807_subset__psubset__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_set_set_nat @ B2 @ C )
=> ( ord_less_set_set_nat @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_808_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_809_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_810_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_811_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_812_not__psubset__empty,axiom,
! [A2: set_set_nat] :
~ ( ord_less_set_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% not_psubset_empty
thf(fact_813_less__supI1,axiom,
! [X2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ A )
=> ( ord_less_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_814_less__supI1,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X2 @ A )
=> ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_815_less__supI2,axiom,
! [X2: set_set_nat,B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ B )
=> ( ord_less_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_816_less__supI2,axiom,
! [X2: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X2 @ B )
=> ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_817_sup_Oabsorb3,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ( sup_sup_set_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_818_sup_Oabsorb3,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_819_sup_Oabsorb4,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( sup_sup_set_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_820_sup_Oabsorb4,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_821_sup_Ostrict__boundedE,axiom,
! [B: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ ( sup_sup_set_set_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_set_set_nat @ B @ A )
=> ~ ( ord_less_set_set_nat @ C2 @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_822_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ C2 @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_823_sup_Ostrict__order__iff,axiom,
( ord_less_set_set_nat
= ( ^ [B5: set_set_nat,A4: set_set_nat] :
( ( A4
= ( sup_sup_set_set_nat @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_824_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( A4
= ( sup_sup_set_nat @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_825_sup_Ostrict__coboundedI1,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ C2 @ A )
=> ( ord_less_set_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_826_sup_Ostrict__coboundedI1,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C2 @ A )
=> ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_827_sup_Ostrict__coboundedI2,axiom,
! [C2: set_set_nat,B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ C2 @ B )
=> ( ord_less_set_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_828_sup_Ostrict__coboundedI2,axiom,
! [C2: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C2 @ B )
=> ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_829_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B7: nat] : ( member_nat @ B7 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_830_psubset__imp__ex__mem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ? [B7: set_nat] : ( member_set_nat @ B7 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_831_algebra__single__set,axiom,
! [X4: set_set_nat,S: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ S )
=> ( sigma_5697435980195335136et_nat @ S @ ( insert_set_set_nat @ bot_bot_set_set_nat @ ( insert_set_set_nat @ X4 @ ( insert_set_set_nat @ ( minus_2163939370556025621et_nat @ S @ X4 ) @ ( insert_set_set_nat @ S @ bot_bo7198184520161983622et_nat ) ) ) ) ) ) ).
% algebra_single_set
thf(fact_832_algebra__single__set,axiom,
! [X4: set_nat,S: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ S )
=> ( sigma_algebra_nat @ S @ ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ X4 @ ( insert_set_nat @ ( minus_minus_set_nat @ S @ X4 ) @ ( insert_set_nat @ S @ bot_bot_set_set_nat ) ) ) ) ) ) ).
% algebra_single_set
thf(fact_833_ring__of__setsI,axiom,
! [M: set_set_set_nat,Omega2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ M @ ( pow_set_nat @ Omega2 ) )
=> ( ( member_set_set_nat @ bot_bot_set_set_nat @ M )
=> ( ! [A6: set_set_nat,B7: set_set_nat] :
( ( member_set_set_nat @ A6 @ M )
=> ( ( member_set_set_nat @ B7 @ M )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ A6 @ B7 ) @ M ) ) )
=> ( ! [A6: set_set_nat,B7: set_set_nat] :
( ( member_set_set_nat @ A6 @ M )
=> ( ( member_set_set_nat @ B7 @ M )
=> ( member_set_set_nat @ ( minus_2163939370556025621et_nat @ A6 @ B7 ) @ M ) ) )
=> ( sigma_5512005348123348494et_nat @ Omega2 @ M ) ) ) ) ) ).
% ring_of_setsI
thf(fact_834_ring__of__setsI,axiom,
! [M: set_set_nat,Omega2: set_nat] :
( ( ord_le6893508408891458716et_nat @ M @ ( pow_nat @ Omega2 ) )
=> ( ( member_set_nat @ bot_bot_set_nat @ M )
=> ( ! [A6: set_nat,B7: set_nat] :
( ( member_set_nat @ A6 @ M )
=> ( ( member_set_nat @ B7 @ M )
=> ( member_set_nat @ ( sup_sup_set_nat @ A6 @ B7 ) @ M ) ) )
=> ( ! [A6: set_nat,B7: set_nat] :
( ( member_set_nat @ A6 @ M )
=> ( ( member_set_nat @ B7 @ M )
=> ( member_set_nat @ ( minus_minus_set_nat @ A6 @ B7 ) @ M ) ) )
=> ( sigma_8325262026724180568ts_nat @ Omega2 @ M ) ) ) ) ) ).
% ring_of_setsI
thf(fact_835_ring__of__sets__iff,axiom,
( sigma_5512005348123348494et_nat
= ( ^ [Omega: set_set_nat,M2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ M2 @ ( pow_set_nat @ Omega ) )
& ( member_set_set_nat @ bot_bot_set_set_nat @ M2 )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ M2 )
=> ! [Y4: set_set_nat] :
( ( member_set_set_nat @ Y4 @ M2 )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ X3 @ Y4 ) @ M2 ) ) )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ M2 )
=> ! [Y4: set_set_nat] :
( ( member_set_set_nat @ Y4 @ M2 )
=> ( member_set_set_nat @ ( minus_2163939370556025621et_nat @ X3 @ Y4 ) @ M2 ) ) ) ) ) ) ).
% ring_of_sets_iff
thf(fact_836_ring__of__sets__iff,axiom,
( sigma_8325262026724180568ts_nat
= ( ^ [Omega: set_nat,M2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ M2 @ ( pow_nat @ Omega ) )
& ( member_set_nat @ bot_bot_set_nat @ M2 )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ M2 )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ M2 )
=> ( member_set_nat @ ( sup_sup_set_nat @ X3 @ Y4 ) @ M2 ) ) )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ M2 )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ M2 )
=> ( member_set_nat @ ( minus_minus_set_nat @ X3 @ Y4 ) @ M2 ) ) ) ) ) ) ).
% ring_of_sets_iff
thf(fact_837_sigma__algebra__single__set,axiom,
! [X4: set_set_nat,S: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ S )
=> ( sigma_3186688988589110745et_nat @ S @ ( insert_set_set_nat @ bot_bot_set_set_nat @ ( insert_set_set_nat @ X4 @ ( insert_set_set_nat @ ( minus_2163939370556025621et_nat @ S @ X4 ) @ ( insert_set_set_nat @ S @ bot_bo7198184520161983622et_nat ) ) ) ) ) ) ).
% sigma_algebra_single_set
thf(fact_838_sigma__algebra__single__set,axiom,
! [X4: set_nat,S: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ S )
=> ( sigma_8817008012692346403ra_nat @ S @ ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ X4 @ ( insert_set_nat @ ( minus_minus_set_nat @ S @ X4 ) @ ( insert_set_nat @ S @ bot_bot_set_set_nat ) ) ) ) ) ) ).
% sigma_algebra_single_set
thf(fact_839_chains__extend,axiom,
! [C2: set_set_set_nat,S: set_set_set_nat,Z: set_set_nat] :
( ( member2946998982187404937et_nat @ C2 @ ( chains_set_nat @ S ) )
=> ( ( member_set_set_nat @ Z @ S )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ C2 )
=> ( ord_le6893508408891458716et_nat @ X @ Z ) )
=> ( member2946998982187404937et_nat @ ( sup_su4213647025997063966et_nat @ ( insert_set_set_nat @ Z @ bot_bo7198184520161983622et_nat ) @ C2 ) @ ( chains_set_nat @ S ) ) ) ) ) ).
% chains_extend
thf(fact_840_chains__extend,axiom,
! [C2: set_set_nat,S: set_set_nat,Z: set_nat] :
( ( member_set_set_nat @ C2 @ ( chains_nat @ S ) )
=> ( ( member_set_nat @ Z @ S )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ C2 )
=> ( ord_less_eq_set_nat @ X @ Z ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ ( insert_set_nat @ Z @ bot_bot_set_set_nat ) @ C2 ) @ ( chains_nat @ S ) ) ) ) ) ).
% chains_extend
thf(fact_841_chainsD2,axiom,
! [C2: set_set_nat,S: set_set_nat] :
( ( member_set_set_nat @ C2 @ ( chains_nat @ S ) )
=> ( ord_le6893508408891458716et_nat @ C2 @ S ) ) ).
% chainsD2
thf(fact_842_Zorn__Lemma2,axiom,
! [A2: set_set_set_nat] :
( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ ( chains_set_nat @ A2 ) )
=> ? [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
& ! [Xb: set_set_nat] :
( ( member_set_set_nat @ Xb @ X )
=> ( ord_le6893508408891458716et_nat @ Xb @ Xa ) ) ) )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_843_Zorn__Lemma2,axiom,
! [A2: set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ ( chains_nat @ A2 ) )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
& ! [Xb: set_nat] :
( ( member_set_nat @ Xb @ X )
=> ( ord_less_eq_set_nat @ Xb @ Xa ) ) ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_844_chainsD,axiom,
! [C2: set_set_set_nat,S: set_set_set_nat,X2: set_set_nat,Y2: set_set_nat] :
( ( member2946998982187404937et_nat @ C2 @ ( chains_set_nat @ S ) )
=> ( ( member_set_set_nat @ X2 @ C2 )
=> ( ( member_set_set_nat @ Y2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
| ( ord_le6893508408891458716et_nat @ Y2 @ X2 ) ) ) ) ) ).
% chainsD
thf(fact_845_chainsD,axiom,
! [C2: set_set_nat,S: set_set_nat,X2: set_nat,Y2: set_nat] :
( ( member_set_set_nat @ C2 @ ( chains_nat @ S ) )
=> ( ( member_set_nat @ X2 @ C2 )
=> ( ( member_set_nat @ Y2 @ C2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
| ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ) ).
% chainsD
thf(fact_846_ring__of__sets_OUn,axiom,
! [Omega2: set_set_nat,M: set_set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( sigma_5512005348123348494et_nat @ Omega2 @ M )
=> ( ( member_set_set_nat @ A @ M )
=> ( ( member_set_set_nat @ B @ M )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ M ) ) ) ) ).
% ring_of_sets.Un
thf(fact_847_ring__of__sets_OUn,axiom,
! [Omega2: set_nat,M: set_set_nat,A: set_nat,B: set_nat] :
( ( sigma_8325262026724180568ts_nat @ Omega2 @ M )
=> ( ( member_set_nat @ A @ M )
=> ( ( member_set_nat @ B @ M )
=> ( member_set_nat @ ( sup_sup_set_nat @ A @ B ) @ M ) ) ) ) ).
% ring_of_sets.Un
thf(fact_848_ring__of__sets_Oinsert__in__sets,axiom,
! [Omega2: set_nat,M: set_set_nat,X2: nat,A2: set_nat] :
( ( sigma_8325262026724180568ts_nat @ Omega2 @ M )
=> ( ( member_set_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) @ M )
=> ( ( member_set_nat @ A2 @ M )
=> ( member_set_nat @ ( insert_nat @ X2 @ A2 ) @ M ) ) ) ) ).
% ring_of_sets.insert_in_sets
thf(fact_849_ring__of__sets_Oinsert__in__sets,axiom,
! [Omega2: set_set_nat,M: set_set_set_nat,X2: set_nat,A2: set_set_nat] :
( ( sigma_5512005348123348494et_nat @ Omega2 @ M )
=> ( ( member_set_set_nat @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) @ M )
=> ( ( member_set_set_nat @ A2 @ M )
=> ( member_set_set_nat @ ( insert_set_nat @ X2 @ A2 ) @ M ) ) ) ) ).
% ring_of_sets.insert_in_sets
thf(fact_850_sigma__algebra__trivial,axiom,
! [Omega2: set_nat] : ( sigma_8817008012692346403ra_nat @ Omega2 @ ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ Omega2 @ bot_bot_set_set_nat ) ) ) ).
% sigma_algebra_trivial
thf(fact_851_sigma__algebra__trivial,axiom,
! [Omega2: set_set_nat] : ( sigma_3186688988589110745et_nat @ Omega2 @ ( insert_set_set_nat @ bot_bot_set_set_nat @ ( insert_set_set_nat @ Omega2 @ bot_bo7198184520161983622et_nat ) ) ) ).
% sigma_algebra_trivial
thf(fact_852_closed__cdi__Un,axiom,
! [Omega2: set_nat,M: set_set_nat,A2: set_nat,B2: set_nat] :
( ( sigma_closed_cdi_nat @ Omega2 @ M )
=> ( ( member_set_nat @ bot_bot_set_nat @ M )
=> ( ( member_set_nat @ A2 @ M )
=> ( ( member_set_nat @ B2 @ M )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( member_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ M ) ) ) ) ) ) ).
% closed_cdi_Un
thf(fact_853_closed__cdi__Un,axiom,
! [Omega2: set_set_nat,M: set_set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( sigma_3476109907249799476et_nat @ Omega2 @ M )
=> ( ( member_set_set_nat @ bot_bot_set_set_nat @ M )
=> ( ( member_set_set_nat @ A2 @ M )
=> ( ( member_set_set_nat @ B2 @ M )
=> ( ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ M ) ) ) ) ) ) ).
% closed_cdi_Un
thf(fact_854_sigma__sets__singleton,axiom,
! [X4: set_set_nat,S: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ S )
=> ( ( sigma_5025102979728185774et_nat @ S @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) )
= ( insert_set_set_nat @ bot_bot_set_set_nat @ ( insert_set_set_nat @ X4 @ ( insert_set_set_nat @ ( minus_2163939370556025621et_nat @ S @ X4 ) @ ( insert_set_set_nat @ S @ bot_bo7198184520161983622et_nat ) ) ) ) ) ) ).
% sigma_sets_singleton
thf(fact_855_sigma__sets__singleton,axiom,
! [X4: set_nat,S: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ S )
=> ( ( sigma_sigma_sets_nat @ S @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
= ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ X4 @ ( insert_set_nat @ ( minus_minus_set_nat @ S @ X4 ) @ ( insert_set_nat @ S @ bot_bot_set_set_nat ) ) ) ) ) ) ).
% sigma_sets_singleton
thf(fact_856_algebra_Osmallest__ccdi__sets__Un,axiom,
! [Omega2: set_nat,M: set_set_nat,A2: set_nat,B2: set_nat] :
( ( sigma_algebra_nat @ Omega2 @ M )
=> ( ( member_set_nat @ A2 @ ( sigma_5553761350045521333ts_nat @ Omega2 @ M ) )
=> ( ( member_set_nat @ B2 @ ( sigma_5553761350045521333ts_nat @ Omega2 @ M ) )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( member_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sigma_5553761350045521333ts_nat @ Omega2 @ M ) ) ) ) ) ) ).
% algebra.smallest_ccdi_sets_Un
thf(fact_857_algebra_Osmallest__ccdi__sets__Un,axiom,
! [Omega2: set_set_nat,M: set_set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( sigma_5697435980195335136et_nat @ Omega2 @ M )
=> ( ( member_set_set_nat @ A2 @ ( sigma_1895591208183295339et_nat @ Omega2 @ M ) )
=> ( ( member_set_set_nat @ B2 @ ( sigma_1895591208183295339et_nat @ Omega2 @ M ) )
=> ( ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ ( sigma_1895591208183295339et_nat @ Omega2 @ M ) ) ) ) ) ) ).
% algebra.smallest_ccdi_sets_Un
thf(fact_858_subset_Ochain__extend,axiom,
! [A2: set_set_nat,C: set_set_nat,Z: set_nat] :
( ( pred_chain_set_nat @ A2 @ ord_less_set_nat @ C )
=> ( ( member_set_nat @ Z @ A2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ C )
=> ( sup_su3254969269353549134_nat_o @ ord_less_set_nat
@ ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 )
@ X
@ Z ) )
=> ( pred_chain_set_nat @ A2 @ ord_less_set_nat @ ( sup_sup_set_set_nat @ ( insert_set_nat @ Z @ bot_bot_set_set_nat ) @ C ) ) ) ) ) ).
% subset.chain_extend
thf(fact_859_sigma__sets__single,axiom,
! [A2: set_set_nat] :
( ( sigma_5025102979728185774et_nat @ A2 @ ( insert_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) )
= ( insert_set_set_nat @ bot_bot_set_set_nat @ ( insert_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ) ) ).
% sigma_sets_single
thf(fact_860_sigma__sets__single,axiom,
! [A2: set_nat] :
( ( sigma_sigma_sets_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
= ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ) ).
% sigma_sets_single
thf(fact_861_pred__on_Ochain__def,axiom,
( pred_chain_set_nat
= ( ^ [A3: set_set_nat,P3: set_nat > set_nat > $o,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A3 )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C3 )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ C3 )
=> ( ( sup_su3254969269353549134_nat_o @ P3
@ ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 )
@ X3
@ Y4 )
| ( sup_su3254969269353549134_nat_o @ P3
@ ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 )
@ Y4
@ X3 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_862_pred__on_Ochain__def,axiom,
( pred_chain_nat
= ( ^ [A3: set_nat,P3: nat > nat > $o,C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A3 )
& ! [X3: nat] :
( ( member_nat @ X3 @ C3 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ C3 )
=> ( ( sup_sup_nat_nat_o @ P3
@ ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 )
@ X3
@ Y4 )
| ( sup_sup_nat_nat_o @ P3
@ ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 )
@ Y4
@ X3 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_863_pred__on_OchainI,axiom,
! [C: set_set_nat,A2: set_set_nat,P2: set_nat > set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( member_set_nat @ X @ C )
=> ( ( member_set_nat @ Y @ C )
=> ( ( sup_su3254969269353549134_nat_o @ P2
@ ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 )
@ X
@ Y )
| ( sup_su3254969269353549134_nat_o @ P2
@ ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 )
@ Y
@ X ) ) ) )
=> ( pred_chain_set_nat @ A2 @ P2 @ C ) ) ) ).
% pred_on.chainI
thf(fact_864_pred__on_OchainI,axiom,
! [C: set_nat,A2: set_nat,P2: nat > nat > $o] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ! [X: nat,Y: nat] :
( ( member_nat @ X @ C )
=> ( ( member_nat @ Y @ C )
=> ( ( sup_sup_nat_nat_o @ P2
@ ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 )
@ X
@ Y )
| ( sup_sup_nat_nat_o @ P2
@ ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 )
@ Y
@ X ) ) ) )
=> ( pred_chain_nat @ A2 @ P2 @ C ) ) ) ).
% pred_on.chainI
thf(fact_865_pred__on_Ochain__empty,axiom,
! [A2: set_nat,P2: nat > nat > $o] : ( pred_chain_nat @ A2 @ P2 @ bot_bot_set_nat ) ).
% pred_on.chain_empty
thf(fact_866_pred__on_Ochain__empty,axiom,
! [A2: set_set_nat,P2: set_nat > set_nat > $o] : ( pred_chain_set_nat @ A2 @ P2 @ bot_bot_set_set_nat ) ).
% pred_on.chain_empty
thf(fact_867_sigma__sets__Un,axiom,
! [A: set_set_nat,Sp: set_set_nat,A2: set_set_set_nat,B: set_set_nat] :
( ( member_set_set_nat @ A @ ( sigma_5025102979728185774et_nat @ Sp @ A2 ) )
=> ( ( member_set_set_nat @ B @ ( sigma_5025102979728185774et_nat @ Sp @ A2 ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sigma_5025102979728185774et_nat @ Sp @ A2 ) ) ) ) ).
% sigma_sets_Un
thf(fact_868_sigma__sets__Un,axiom,
! [A: set_nat,Sp: set_nat,A2: set_set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( sigma_sigma_sets_nat @ Sp @ A2 ) )
=> ( ( member_set_nat @ B @ ( sigma_sigma_sets_nat @ Sp @ A2 ) )
=> ( member_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sigma_sigma_sets_nat @ Sp @ A2 ) ) ) ) ).
% sigma_sets_Un
thf(fact_869_sigma__sets__superset__generator,axiom,
! [A2: set_set_nat,X4: set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ ( sigma_sigma_sets_nat @ X4 @ A2 ) ) ).
% sigma_sets_superset_generator
thf(fact_870_sigma__sets__subseteq,axiom,
! [A2: set_set_nat,B2: set_set_nat,X4: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ X4 @ A2 ) @ ( sigma_sigma_sets_nat @ X4 @ B2 ) ) ) ).
% sigma_sets_subseteq
thf(fact_871_sigma__sets__mono,axiom,
! [A2: set_set_nat,X4: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( sigma_sigma_sets_nat @ X4 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ X4 @ A2 ) @ ( sigma_sigma_sets_nat @ X4 @ B2 ) ) ) ).
% sigma_sets_mono
thf(fact_872_sigma__sets_OEmpty,axiom,
! [Sp: set_nat,A2: set_set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( sigma_sigma_sets_nat @ Sp @ A2 ) ) ).
% sigma_sets.Empty
thf(fact_873_sigma__sets_OEmpty,axiom,
! [Sp: set_set_nat,A2: set_set_set_nat] : ( member_set_set_nat @ bot_bot_set_set_nat @ ( sigma_5025102979728185774et_nat @ Sp @ A2 ) ) ).
% sigma_sets.Empty
thf(fact_874_algebra_Osigma__property__disjoint__lemma,axiom,
! [Omega2: set_nat,M: set_set_nat,C: set_set_nat] :
( ( sigma_algebra_nat @ Omega2 @ M )
=> ( ( ord_le6893508408891458716et_nat @ M @ C )
=> ( ( sigma_closed_cdi_nat @ Omega2 @ C )
=> ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ Omega2 @ M ) @ C ) ) ) ) ).
% algebra.sigma_property_disjoint_lemma
thf(fact_875_sigma__sets__sigma__sets__eq,axiom,
! [M: set_set_nat,S: set_nat] :
( ( ord_le6893508408891458716et_nat @ M @ ( pow_nat @ S ) )
=> ( ( sigma_sigma_sets_nat @ S @ ( sigma_sigma_sets_nat @ S @ M ) )
= ( sigma_sigma_sets_nat @ S @ M ) ) ) ).
% sigma_sets_sigma_sets_eq
thf(fact_876_sigma__sets__mono_H_H,axiom,
! [A2: set_nat,C: set_nat,D2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A2 @ ( sigma_sigma_sets_nat @ C @ D2 ) )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D2 )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ ( pow_nat @ C ) )
=> ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ A2 @ B2 ) @ ( sigma_sigma_sets_nat @ C @ D2 ) ) ) ) ) ).
% sigma_sets_mono''
thf(fact_877_subset__Zorn,axiom,
! [A2: set_set_set_nat] :
( ! [C4: set_set_set_nat] :
( ( pred_c5700569349699901905et_nat @ A2 @ ord_less_set_set_nat @ C4 )
=> ? [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ A2 )
& ! [Xa2: set_set_nat] :
( ( member_set_set_nat @ Xa2 @ C4 )
=> ( ord_le6893508408891458716et_nat @ Xa2 @ X6 ) ) ) )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% subset_Zorn
thf(fact_878_subset__Zorn,axiom,
! [A2: set_set_nat] :
( ! [C4: set_set_nat] :
( ( pred_chain_set_nat @ A2 @ ord_less_set_nat @ C4 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A2 )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ C4 )
=> ( ord_less_eq_set_nat @ Xa2 @ X6 ) ) ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% subset_Zorn
thf(fact_879_subset_Ochain__def,axiom,
! [A2: set_set_nat,C: set_set_nat] :
( ( pred_chain_set_nat @ A2 @ ord_less_set_nat @ C )
= ( ( ord_le6893508408891458716et_nat @ C @ A2 )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ C )
=> ( ( sup_su3254969269353549134_nat_o @ ord_less_set_nat
@ ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 )
@ X3
@ Y4 )
| ( sup_su3254969269353549134_nat_o @ ord_less_set_nat
@ ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 )
@ Y4
@ X3 ) ) ) ) ) ) ).
% subset.chain_def
thf(fact_880_subset_OchainI,axiom,
! [C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ! [X: set_nat,Y: set_nat] :
( ( member_set_nat @ X @ C )
=> ( ( member_set_nat @ Y @ C )
=> ( ( sup_su3254969269353549134_nat_o @ ord_less_set_nat
@ ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 )
@ X
@ Y )
| ( sup_su3254969269353549134_nat_o @ ord_less_set_nat
@ ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 )
@ Y
@ X ) ) ) )
=> ( pred_chain_set_nat @ A2 @ ord_less_set_nat @ C ) ) ) ).
% subset.chainI
thf(fact_881_sigma__algebra_Osigma__sets__subset,axiom,
! [Omega2: set_nat,M: set_set_nat,A: set_set_nat] :
( ( sigma_8817008012692346403ra_nat @ Omega2 @ M )
=> ( ( ord_le6893508408891458716et_nat @ A @ M )
=> ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ Omega2 @ A ) @ M ) ) ) ).
% sigma_algebra.sigma_sets_subset
thf(fact_882_sigma__algebra_Osigma__sets__subset_H,axiom,
! [Omega2: set_nat,M: set_set_nat,A: set_set_nat,Omega3: set_nat] :
( ( sigma_8817008012692346403ra_nat @ Omega2 @ M )
=> ( ( ord_le6893508408891458716et_nat @ A @ M )
=> ( ( member_set_nat @ Omega3 @ M )
=> ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ Omega3 @ A ) @ M ) ) ) ) ).
% sigma_algebra.sigma_sets_subset'
thf(fact_883_sigma__sets__into__sp,axiom,
! [A2: set_set_set_nat,Sp: set_set_nat,X2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( pow_set_nat @ Sp ) )
=> ( ( member_set_set_nat @ X2 @ ( sigma_5025102979728185774et_nat @ Sp @ A2 ) )
=> ( ord_le6893508408891458716et_nat @ X2 @ Sp ) ) ) ).
% sigma_sets_into_sp
thf(fact_884_sigma__sets__into__sp,axiom,
! [A2: set_set_nat,Sp: set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( pow_nat @ Sp ) )
=> ( ( member_set_nat @ X2 @ ( sigma_sigma_sets_nat @ Sp @ A2 ) )
=> ( ord_less_eq_set_nat @ X2 @ Sp ) ) ) ).
% sigma_sets_into_sp
thf(fact_885_sigma__sets__empty__eq,axiom,
! [A2: set_set_nat] :
( ( sigma_5025102979728185774et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( insert_set_set_nat @ bot_bot_set_set_nat @ ( insert_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ) ) ).
% sigma_sets_empty_eq
thf(fact_886_sigma__sets__empty__eq,axiom,
! [A2: set_nat] :
( ( sigma_sigma_sets_nat @ A2 @ bot_bot_set_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ) ).
% sigma_sets_empty_eq
thf(fact_887_subset__chain__def,axiom,
! [A7: set_set_set_nat,C5: set_set_set_nat] :
( ( pred_c5700569349699901905et_nat @ A7 @ ord_less_set_set_nat @ C5 )
= ( ( ord_le9131159989063066194et_nat @ C5 @ A7 )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ C5 )
=> ! [Y4: set_set_nat] :
( ( member_set_set_nat @ Y4 @ C5 )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
| ( ord_le6893508408891458716et_nat @ Y4 @ X3 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_888_subset__chain__def,axiom,
! [A7: set_set_nat,C5: set_set_nat] :
( ( pred_chain_set_nat @ A7 @ ord_less_set_nat @ C5 )
= ( ( ord_le6893508408891458716et_nat @ C5 @ A7 )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C5 )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ C5 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y4 )
| ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_889_sigma__algebra__sigma__sets,axiom,
! [A: set_set_nat,Omega2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( pow_nat @ Omega2 ) )
=> ( sigma_8817008012692346403ra_nat @ Omega2 @ ( sigma_sigma_sets_nat @ Omega2 @ A ) ) ) ).
% sigma_algebra_sigma_sets
thf(fact_890_subset__chain__insert,axiom,
! [A7: set_set_set_nat,B2: set_set_nat,B8: set_set_set_nat] :
( ( pred_c5700569349699901905et_nat @ A7 @ ord_less_set_set_nat @ ( insert_set_set_nat @ B2 @ B8 ) )
= ( ( member_set_set_nat @ B2 @ A7 )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B8 )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ B2 )
| ( ord_le6893508408891458716et_nat @ B2 @ X3 ) ) )
& ( pred_c5700569349699901905et_nat @ A7 @ ord_less_set_set_nat @ B8 ) ) ) ).
% subset_chain_insert
thf(fact_891_subset__chain__insert,axiom,
! [A7: set_set_nat,B2: set_nat,B8: set_set_nat] :
( ( pred_chain_set_nat @ A7 @ ord_less_set_nat @ ( insert_set_nat @ B2 @ B8 ) )
= ( ( member_set_nat @ B2 @ A7 )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B8 )
=> ( ( ord_less_eq_set_nat @ X3 @ B2 )
| ( ord_less_eq_set_nat @ B2 @ X3 ) ) )
& ( pred_chain_set_nat @ A7 @ ord_less_set_nat @ B8 ) ) ) ).
% subset_chain_insert
thf(fact_892_pred__on_Ochain__extend,axiom,
! [A2: set_nat,P2: nat > nat > $o,C: set_nat,Z: nat] :
( ( pred_chain_nat @ A2 @ P2 @ C )
=> ( ( member_nat @ Z @ A2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ C )
=> ( sup_sup_nat_nat_o @ P2
@ ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 )
@ X
@ Z ) )
=> ( pred_chain_nat @ A2 @ P2 @ ( sup_sup_set_nat @ ( insert_nat @ Z @ bot_bot_set_nat ) @ C ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_893_pred__on_Ochain__extend,axiom,
! [A2: set_set_nat,P2: set_nat > set_nat > $o,C: set_set_nat,Z: set_nat] :
( ( pred_chain_set_nat @ A2 @ P2 @ C )
=> ( ( member_set_nat @ Z @ A2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ C )
=> ( sup_su3254969269353549134_nat_o @ P2
@ ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 )
@ X
@ Z ) )
=> ( pred_chain_set_nat @ A2 @ P2 @ ( sup_sup_set_set_nat @ ( insert_set_nat @ Z @ bot_bot_set_set_nat ) @ C ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_894_closed__cdi__subset,axiom,
! [Omega2: set_nat,M: set_set_nat] :
( ( sigma_closed_cdi_nat @ Omega2 @ M )
=> ( ord_le6893508408891458716et_nat @ M @ ( pow_nat @ Omega2 ) ) ) ).
% closed_cdi_subset
thf(fact_895_subset__class_Osmallest__ccdi__sets,axiom,
! [Omega2: set_nat,M: set_set_nat] :
( ( sigma_9101811122110323416ss_nat @ Omega2 @ M )
=> ( ord_le6893508408891458716et_nat @ ( sigma_5553761350045521333ts_nat @ Omega2 @ M ) @ ( pow_nat @ Omega2 ) ) ) ).
% subset_class.smallest_ccdi_sets
thf(fact_896_subset__Zorn__nonempty,axiom,
! [A7: set_set_set_nat] :
( ( A7 != bot_bo7198184520161983622et_nat )
=> ( ! [C6: set_set_set_nat] :
( ( C6 != bot_bo7198184520161983622et_nat )
=> ( ( pred_c5700569349699901905et_nat @ A7 @ ord_less_set_set_nat @ C6 )
=> ( member_set_set_nat @ ( comple548664676211718543et_nat @ C6 ) @ A7 ) ) )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A7 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A7 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_897_subset__Zorn__nonempty,axiom,
! [A7: set_set_nat] :
( ( A7 != bot_bot_set_set_nat )
=> ( ! [C6: set_set_nat] :
( ( C6 != bot_bot_set_set_nat )
=> ( ( pred_chain_set_nat @ A7 @ ord_less_set_nat @ C6 )
=> ( member_set_nat @ ( comple7399068483239264473et_nat @ C6 ) @ A7 ) ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A7 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A7 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_898_cSup__singleton,axiom,
! [X2: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% cSup_singleton
thf(fact_899_cSup__singleton,axiom,
! [X2: set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= X2 ) ).
% cSup_singleton
thf(fact_900_cSup__eq__maximum,axiom,
! [Z: nat,X4: set_nat] :
( ( member_nat @ Z @ X4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ X4 )
=> ( ord_less_eq_nat @ X @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_901_cSup__eq__maximum,axiom,
! [Z: set_set_nat,X4: set_set_set_nat] :
( ( member_set_set_nat @ Z @ X4 )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ X4 )
=> ( ord_le6893508408891458716et_nat @ X @ Z ) )
=> ( ( comple548664676211718543et_nat @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_902_cSup__eq__maximum,axiom,
! [Z: set_nat,X4: set_set_nat] :
( ( member_set_nat @ Z @ X4 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X4 )
=> ( ord_less_eq_set_nat @ X @ Z ) )
=> ( ( comple7399068483239264473et_nat @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_903_subset__class_Osets__into__space,axiom,
! [Omega2: set_set_nat,M: set_set_set_nat,X2: set_set_nat] :
( ( sigma_1454238057069045390et_nat @ Omega2 @ M )
=> ( ( member_set_set_nat @ X2 @ M )
=> ( ord_le6893508408891458716et_nat @ X2 @ Omega2 ) ) ) ).
% subset_class.sets_into_space
thf(fact_904_subset__class_Osets__into__space,axiom,
! [Omega2: set_nat,M: set_set_nat,X2: set_nat] :
( ( sigma_9101811122110323416ss_nat @ Omega2 @ M )
=> ( ( member_set_nat @ X2 @ M )
=> ( ord_less_eq_set_nat @ X2 @ Omega2 ) ) ) ).
% subset_class.sets_into_space
thf(fact_905_cSup__eq__non__empty,axiom,
! [X4: set_nat,A: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X4 )
=> ( ord_less_eq_nat @ X @ A ) )
=> ( ! [Y: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ X4 )
=> ( ord_less_eq_nat @ X6 @ Y ) )
=> ( ord_less_eq_nat @ A @ Y ) )
=> ( ( complete_Sup_Sup_nat @ X4 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_906_cSup__eq__non__empty,axiom,
! [X4: set_set_set_nat,A: set_set_nat] :
( ( X4 != bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ X4 )
=> ( ord_le6893508408891458716et_nat @ X @ A ) )
=> ( ! [Y: set_set_nat] :
( ! [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ X4 )
=> ( ord_le6893508408891458716et_nat @ X6 @ Y ) )
=> ( ord_le6893508408891458716et_nat @ A @ Y ) )
=> ( ( comple548664676211718543et_nat @ X4 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_907_cSup__eq__non__empty,axiom,
! [X4: set_set_nat,A: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X4 )
=> ( ord_less_eq_set_nat @ X @ A ) )
=> ( ! [Y: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ X4 )
=> ( ord_less_eq_set_nat @ X6 @ Y ) )
=> ( ord_less_eq_set_nat @ A @ Y ) )
=> ( ( comple7399068483239264473et_nat @ X4 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_908_cSup__least,axiom,
! [X4: set_nat,Z: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X4 )
=> ( ord_less_eq_nat @ X @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_909_cSup__least,axiom,
! [X4: set_set_set_nat,Z: set_set_nat] :
( ( X4 != bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ X4 )
=> ( ord_le6893508408891458716et_nat @ X @ Z ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_910_cSup__least,axiom,
! [X4: set_set_nat,Z: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X4 )
=> ( ord_less_eq_set_nat @ X @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_911_less__cSupD,axiom,
! [X4: set_nat,Z: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z @ ( complete_Sup_Sup_nat @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ X4 )
& ( ord_less_nat @ Z @ X ) ) ) ) ).
% less_cSupD
thf(fact_912_less__cSupE,axiom,
! [Y2: nat,X4: set_nat] :
( ( ord_less_nat @ Y2 @ ( complete_Sup_Sup_nat @ X4 ) )
=> ( ( X4 != bot_bot_set_nat )
=> ~ ! [X: nat] :
( ( member_nat @ X @ X4 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ) ) ).
% less_cSupE
thf(fact_913_Zorn__Lemma,axiom,
! [A2: set_set_set_nat] :
( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ ( chains_set_nat @ A2 ) )
=> ( member_set_set_nat @ ( comple548664676211718543et_nat @ X ) @ A2 ) )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_914_Zorn__Lemma,axiom,
! [A2: set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ ( chains_nat @ A2 ) )
=> ( member_set_nat @ ( comple7399068483239264473et_nat @ X ) @ A2 ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_915_subset__class_Ointro,axiom,
! [M: set_set_nat,Omega2: set_nat] :
( ( ord_le6893508408891458716et_nat @ M @ ( pow_nat @ Omega2 ) )
=> ( sigma_9101811122110323416ss_nat @ Omega2 @ M ) ) ).
% subset_class.intro
thf(fact_916_subset__class_Ospace__closed,axiom,
! [Omega2: set_nat,M: set_set_nat] :
( ( sigma_9101811122110323416ss_nat @ Omega2 @ M )
=> ( ord_le6893508408891458716et_nat @ M @ ( pow_nat @ Omega2 ) ) ) ).
% subset_class.space_closed
thf(fact_917_subset__class__def,axiom,
( sigma_9101811122110323416ss_nat
= ( ^ [Omega: set_nat,M2: set_set_nat] : ( ord_le6893508408891458716et_nat @ M2 @ ( pow_nat @ Omega ) ) ) ) ).
% subset_class_def
thf(fact_918_subset__Zorn_H,axiom,
! [A2: set_set_set_nat] :
( ! [C4: set_set_set_nat] :
( ( pred_c5700569349699901905et_nat @ A2 @ ord_less_set_set_nat @ C4 )
=> ( member_set_set_nat @ ( comple548664676211718543et_nat @ C4 ) @ A2 ) )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% subset_Zorn'
thf(fact_919_subset__Zorn_H,axiom,
! [A2: set_set_nat] :
( ! [C4: set_set_nat] :
( ( pred_chain_set_nat @ A2 @ ord_less_set_nat @ C4 )
=> ( member_set_nat @ ( comple7399068483239264473et_nat @ C4 ) @ A2 ) )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% subset_Zorn'
thf(fact_920_subset__class_Osmallest__closed__cdi1,axiom,
! [Omega2: set_nat,M: set_set_nat] :
( ( sigma_9101811122110323416ss_nat @ Omega2 @ M )
=> ( ord_le6893508408891458716et_nat @ M @ ( sigma_5553761350045521333ts_nat @ Omega2 @ M ) ) ) ).
% subset_class.smallest_closed_cdi1
thf(fact_921_le__sup__lexord,axiom,
! [K: set_set_nat > set_set_nat,A2: set_set_nat,B2: set_set_nat,Ca: set_set_nat,C2: set_set_nat,S2: set_set_nat] :
( ( ( ord_less_set_set_nat @ ( K @ A2 ) @ ( K @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ Ca @ B2 ) )
=> ( ( ( ord_less_set_set_nat @ ( K @ B2 ) @ ( K @ A2 ) )
=> ( ord_le6893508408891458716et_nat @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ Ca @ C2 ) )
=> ( ( ~ ( ord_le6893508408891458716et_nat @ ( K @ B2 ) @ ( K @ A2 ) )
=> ( ~ ( ord_le6893508408891458716et_nat @ ( K @ A2 ) @ ( K @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ Ca @ S2 ) ) )
=> ( ord_le6893508408891458716et_nat @ Ca @ ( measur7844494817220136214et_nat @ A2 @ B2 @ K @ S2 @ C2 ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_922_le__sup__lexord,axiom,
! [K: set_set_nat > set_nat,A2: set_set_nat,B2: set_set_nat,Ca: set_set_nat,C2: set_set_nat,S2: set_set_nat] :
( ( ( ord_less_set_nat @ ( K @ A2 ) @ ( K @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ Ca @ B2 ) )
=> ( ( ( ord_less_set_nat @ ( K @ B2 ) @ ( K @ A2 ) )
=> ( ord_le6893508408891458716et_nat @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ Ca @ C2 ) )
=> ( ( ~ ( ord_less_eq_set_nat @ ( K @ B2 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_set_nat @ ( K @ A2 ) @ ( K @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ Ca @ S2 ) ) )
=> ( ord_le6893508408891458716et_nat @ Ca @ ( measur5433014825823400032et_nat @ A2 @ B2 @ K @ S2 @ C2 ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_923_le__sup__lexord,axiom,
! [K: set_nat > set_set_nat,A2: set_nat,B2: set_nat,Ca: set_nat,C2: set_nat,S2: set_nat] :
( ( ( ord_less_set_set_nat @ ( K @ A2 ) @ ( K @ B2 ) )
=> ( ord_less_eq_set_nat @ Ca @ B2 ) )
=> ( ( ( ord_less_set_set_nat @ ( K @ B2 ) @ ( K @ A2 ) )
=> ( ord_less_eq_set_nat @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B2 ) )
=> ( ord_less_eq_set_nat @ Ca @ C2 ) )
=> ( ( ~ ( ord_le6893508408891458716et_nat @ ( K @ B2 ) @ ( K @ A2 ) )
=> ( ~ ( ord_le6893508408891458716et_nat @ ( K @ A2 ) @ ( K @ B2 ) )
=> ( ord_less_eq_set_nat @ Ca @ S2 ) ) )
=> ( ord_less_eq_set_nat @ Ca @ ( measur6315251617118684256et_nat @ A2 @ B2 @ K @ S2 @ C2 ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_924_le__sup__lexord,axiom,
! [K: set_nat > set_nat,A2: set_nat,B2: set_nat,Ca: set_nat,C2: set_nat,S2: set_nat] :
( ( ( ord_less_set_nat @ ( K @ A2 ) @ ( K @ B2 ) )
=> ( ord_less_eq_set_nat @ Ca @ B2 ) )
=> ( ( ( ord_less_set_nat @ ( K @ B2 ) @ ( K @ A2 ) )
=> ( ord_less_eq_set_nat @ Ca @ A2 ) )
=> ( ( ( ( K @ A2 )
= ( K @ B2 ) )
=> ( ord_less_eq_set_nat @ Ca @ C2 ) )
=> ( ( ~ ( ord_less_eq_set_nat @ ( K @ B2 ) @ ( K @ A2 ) )
=> ( ~ ( ord_less_eq_set_nat @ ( K @ A2 ) @ ( K @ B2 ) )
=> ( ord_less_eq_set_nat @ Ca @ S2 ) ) )
=> ( ord_less_eq_set_nat @ Ca @ ( measur5257060982548205482et_nat @ A2 @ B2 @ K @ S2 @ C2 ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_925_Union__Un__distrib,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( comple548664676211718543et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_926_Union__Un__distrib,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_927_complete__lattice__class_OSup__insert,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( comple548664676211718543et_nat @ ( insert_set_set_nat @ A @ A2 ) )
= ( sup_sup_set_set_nat @ A @ ( comple548664676211718543et_nat @ A2 ) ) ) ).
% complete_lattice_class.Sup_insert
thf(fact_928_complete__lattice__class_OSup__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ A2 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% complete_lattice_class.Sup_insert
thf(fact_929_ccpo__Sup__singleton,axiom,
! [X2: set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= X2 ) ).
% ccpo_Sup_singleton
thf(fact_930_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_931_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_set_nat] :
( ( ( comple548664676211718543et_nat @ A2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_932_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_933_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_set_nat] :
( ( bot_bot_set_set_nat
= ( comple548664676211718543et_nat @ A2 ) )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_934_Sup__empty,axiom,
( ( comple548664676211718543et_nat @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% Sup_empty
thf(fact_935_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_936_Sup__eqI,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ! [Y: set_set_nat] :
( ( member_set_set_nat @ Y @ A2 )
=> ( ord_le6893508408891458716et_nat @ Y @ X2 ) )
=> ( ! [Y: set_set_nat] :
( ! [Z4: set_set_nat] :
( ( member_set_set_nat @ Z4 @ A2 )
=> ( ord_le6893508408891458716et_nat @ Z4 @ Y ) )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y ) )
=> ( ( comple548664676211718543et_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_937_Sup__eqI,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ! [Y: set_nat] :
( ( member_set_nat @ Y @ A2 )
=> ( ord_less_eq_set_nat @ Y @ X2 ) )
=> ( ! [Y: set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ A2 )
=> ( ord_less_eq_set_nat @ Z4 @ Y ) )
=> ( ord_less_eq_set_nat @ X2 @ Y ) )
=> ( ( comple7399068483239264473et_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_938_Sup__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [A6: set_set_nat] :
( ( member_set_set_nat @ A6 @ A2 )
=> ? [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ B2 )
& ( ord_le6893508408891458716et_nat @ A6 @ X6 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_939_Sup__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A2 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ B2 )
& ( ord_less_eq_set_nat @ A6 @ X6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_940_Sup__least,axiom,
! [A2: set_set_set_nat,Z: set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ X @ Z ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_941_Sup__least,axiom,
! [A2: set_set_nat,Z: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_942_Sup__upper,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( comple548664676211718543et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_943_Sup__upper,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_944_Sup__le__iff,axiom,
! [A2: set_set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ B )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_945_Sup__le__iff,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_946_Sup__upper2,axiom,
! [U: set_set_nat,A2: set_set_set_nat,V: set_set_nat] :
( ( member_set_set_nat @ U @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ V @ U )
=> ( ord_le6893508408891458716et_nat @ V @ ( comple548664676211718543et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_947_Sup__upper2,axiom,
! [U: set_nat,A2: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A2 )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_948_Union__least,axiom,
! [A2: set_set_set_nat,C: set_set_nat] :
( ! [X7: set_set_nat] :
( ( member_set_set_nat @ X7 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X7 @ C ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ C ) ) ).
% Union_least
thf(fact_949_Union__least,axiom,
! [A2: set_set_nat,C: set_nat] :
( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ A2 )
=> ( ord_less_eq_set_nat @ X7 @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ C ) ) ).
% Union_least
thf(fact_950_Union__upper,axiom,
! [B2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ ( comple548664676211718543et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_951_Union__upper,axiom,
! [B2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_952_Union__subsetI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ? [Y6: set_set_nat] :
( ( member_set_set_nat @ Y6 @ B2 )
& ( ord_le6893508408891458716et_nat @ X @ Y6 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_953_Union__subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ B2 )
& ( ord_less_eq_set_nat @ X @ Y6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_954_empty__Union__conv,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_955_empty__Union__conv,axiom,
! [A2: set_set_set_nat] :
( ( bot_bot_set_set_nat
= ( comple548664676211718543et_nat @ A2 ) )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_956_Union__empty__conv,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_957_Union__empty__conv,axiom,
! [A2: set_set_set_nat] :
( ( ( comple548664676211718543et_nat @ A2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( X3 = bot_bot_set_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_958_Sup__subset__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_959_Sup__subset__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_960_less__eq__Sup,axiom,
! [A2: set_set_set_nat,U: set_set_nat] :
( ! [V2: set_set_nat] :
( ( member_set_set_nat @ V2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ U @ V2 ) )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ U @ ( comple548664676211718543et_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_961_less__eq__Sup,axiom,
! [A2: set_set_nat,U: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A2 )
=> ( ord_less_eq_set_nat @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_962_Union__disjoint,axiom,
! [C: set_set_nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C ) @ A2 )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C )
=> ( ( inf_inf_set_nat @ X3 @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_963_Union__disjoint,axiom,
! [C: set_set_set_nat,A2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( comple548664676211718543et_nat @ C ) @ A2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ C )
=> ( ( inf_inf_set_set_nat @ X3 @ A2 )
= bot_bot_set_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_964_Sup__union__distrib,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( comple548664676211718543et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_965_Sup__union__distrib,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_966_Union__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_967_Union__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_968_Union__empty,axiom,
( ( comple548664676211718543et_nat @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% Union_empty
thf(fact_969_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_970_subset__Pow__Union,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ ( pow_nat @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% subset_Pow_Union
thf(fact_971_Union__insert,axiom,
! [A: set_set_nat,B2: set_set_set_nat] :
( ( comple548664676211718543et_nat @ ( insert_set_set_nat @ A @ B2 ) )
= ( sup_sup_set_set_nat @ A @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Union_insert
thf(fact_972_Union__insert,axiom,
! [A: set_nat,B2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_insert
thf(fact_973_Sup__inter__less__eq,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) @ ( inf_inf_set_set_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_974_Sup__inter__less__eq,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_975_Union__Int__subset,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) @ ( inf_inf_set_set_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Union_Int_subset
thf(fact_976_Union__Int__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_Int_subset
thf(fact_977_ccSup__empty,axiom,
( ( comple548664676211718543et_nat @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% ccSup_empty
thf(fact_978_ccSup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% ccSup_empty
thf(fact_979_insert__partition,axiom,
! [X2: set_nat,F2: set_set_nat] :
( ~ ( member_set_nat @ X2 @ F2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( insert_set_nat @ X2 @ F2 ) )
=> ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ ( insert_set_nat @ X2 @ F2 ) )
=> ( ( X != Xa2 )
=> ( ( inf_inf_set_nat @ X @ Xa2 )
= bot_bot_set_nat ) ) ) )
=> ( ( inf_inf_set_nat @ X2 @ ( comple7399068483239264473et_nat @ F2 ) )
= bot_bot_set_nat ) ) ) ).
% insert_partition
thf(fact_980_insert__partition,axiom,
! [X2: set_set_nat,F2: set_set_set_nat] :
( ~ ( member_set_set_nat @ X2 @ F2 )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ ( insert_set_set_nat @ X2 @ F2 ) )
=> ! [Xa2: set_set_nat] :
( ( member_set_set_nat @ Xa2 @ ( insert_set_set_nat @ X2 @ F2 ) )
=> ( ( X != Xa2 )
=> ( ( inf_inf_set_set_nat @ X @ Xa2 )
= bot_bot_set_set_nat ) ) ) )
=> ( ( inf_inf_set_set_nat @ X2 @ ( comple548664676211718543et_nat @ F2 ) )
= bot_bot_set_set_nat ) ) ) ).
% insert_partition
thf(fact_981_Sup__inf__eq__bot__iff,axiom,
! [B2: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B2 ) @ A )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
=> ( ( inf_inf_set_nat @ X3 @ A )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_982_Sup__inf__eq__bot__iff,axiom,
! [B2: set_set_set_nat,A: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( comple548664676211718543et_nat @ B2 ) @ A )
= bot_bot_set_set_nat )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B2 )
=> ( ( inf_inf_set_set_nat @ X3 @ A )
= bot_bot_set_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_983_finite__subset__Union__chain,axiom,
! [A2: set_set_nat,B8: set_set_set_nat,A7: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ ( comple548664676211718543et_nat @ B8 ) )
=> ( ( B8 != bot_bo7198184520161983622et_nat )
=> ( ( pred_c5700569349699901905et_nat @ A7 @ ord_less_set_set_nat @ B8 )
=> ~ ! [B4: set_set_nat] :
( ( member_set_set_nat @ B4 @ B8 )
=> ~ ( ord_le6893508408891458716et_nat @ A2 @ B4 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_984_finite__subset__Union__chain,axiom,
! [A2: set_nat,B8: set_set_nat,A7: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B8 ) )
=> ( ( B8 != bot_bot_set_set_nat )
=> ( ( pred_chain_set_nat @ A7 @ ord_less_set_nat @ B8 )
=> ~ ! [B4: set_nat] :
( ( member_set_nat @ B4 @ B8 )
=> ~ ( ord_less_eq_set_nat @ A2 @ B4 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_985_cSup__inter__less__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( ( condit2214826472909112428ve_nat @ A2 )
=> ( ( condit2214826472909112428ve_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A2 ) @ ( complete_Sup_Sup_nat @ B2 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_986_cSup__inter__less__eq,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( condit3670481866171438296et_nat @ A2 )
=> ( ( condit3670481866171438296et_nat @ B2 )
=> ( ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
!= bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) @ ( sup_sup_set_set_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_987_cSup__inter__less__eq,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( condit5477540289124974626et_nat @ A2 )
=> ( ( condit5477540289124974626et_nat @ B2 )
=> ( ( ( inf_inf_set_set_nat @ A2 @ B2 )
!= bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) @ ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_988_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_989_finite__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( insert_set_nat @ A @ A2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ).
% finite_insert
thf(fact_990_finite__Un,axiom,
! [F2: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F2 @ G ) )
= ( ( finite1152437895449049373et_nat @ F2 )
& ( finite1152437895449049373et_nat @ G ) ) ) ).
% finite_Un
thf(fact_991_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_992_bdd__above_OI,axiom,
! [A2: set_nat,M: nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ M ) )
=> ( condit2214826472909112428ve_nat @ A2 ) ) ).
% bdd_above.I
thf(fact_993_bdd__above_OI,axiom,
! [A2: set_set_set_nat,M: set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ X @ M ) )
=> ( condit3670481866171438296et_nat @ A2 ) ) ).
% bdd_above.I
thf(fact_994_bdd__above_OI,axiom,
! [A2: set_set_nat,M: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ M ) )
=> ( condit5477540289124974626et_nat @ A2 ) ) ).
% bdd_above.I
thf(fact_995_bdd__above__empty,axiom,
condit2214826472909112428ve_nat @ bot_bot_set_nat ).
% bdd_above_empty
thf(fact_996_bdd__above__empty,axiom,
condit5477540289124974626et_nat @ bot_bot_set_set_nat ).
% bdd_above_empty
thf(fact_997_bdd__above__insert,axiom,
! [A: nat,A2: set_nat] :
( ( condit2214826472909112428ve_nat @ ( insert_nat @ A @ A2 ) )
= ( condit2214826472909112428ve_nat @ A2 ) ) ).
% bdd_above_insert
thf(fact_998_bdd__above__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( condit5477540289124974626et_nat @ ( insert_set_nat @ A @ A2 ) )
= ( condit5477540289124974626et_nat @ A2 ) ) ).
% bdd_above_insert
thf(fact_999_bdd__above__Un,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( condit5477540289124974626et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( ( condit5477540289124974626et_nat @ A2 )
& ( condit5477540289124974626et_nat @ B2 ) ) ) ).
% bdd_above_Un
thf(fact_1000_bdd__above__Un,axiom,
! [A2: set_nat,B2: set_nat] :
( ( condit2214826472909112428ve_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( condit2214826472909112428ve_nat @ A2 )
& ( condit2214826472909112428ve_nat @ B2 ) ) ) ).
% bdd_above_Un
thf(fact_1001_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1002_finite__Diff__insert,axiom,
! [A2: set_set_nat,A: set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) ) )
= ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1003_rev__finite__subset,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1004_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1005_infinite__super,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_1006_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_1007_finite__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_1008_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_1009_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1010_finite__has__minimal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ( ord_le6893508408891458716et_nat @ X @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1011_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( ord_less_eq_set_nat @ X @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1012_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ A @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1013_finite__has__maximal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ( ord_le6893508408891458716et_nat @ A @ X )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1014_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( ord_less_eq_set_nat @ A @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1015_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1016_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_1017_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1018_infinite__imp__nonempty,axiom,
! [S: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1019_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1020_finite_OinsertI,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite1152437895449049373et_nat @ ( insert_set_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1021_finite__UnI,axiom,
! [F2: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( finite1152437895449049373et_nat @ G )
=> ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_1022_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_1023_Un__infinite,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_1024_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_1025_infinite__Un,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) )
= ( ~ ( finite1152437895449049373et_nat @ S )
| ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_1026_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_1027_cSup__upper,axiom,
! [X2: nat,X4: set_nat] :
( ( member_nat @ X2 @ X4 )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ord_less_eq_nat @ X2 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ).
% cSup_upper
thf(fact_1028_cSup__upper,axiom,
! [X2: set_set_nat,X4: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ X4 )
=> ( ( condit3670481866171438296et_nat @ X4 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( comple548664676211718543et_nat @ X4 ) ) ) ) ).
% cSup_upper
thf(fact_1029_cSup__upper,axiom,
! [X2: set_nat,X4: set_set_nat] :
( ( member_set_nat @ X2 @ X4 )
=> ( ( condit5477540289124974626et_nat @ X4 )
=> ( ord_less_eq_set_nat @ X2 @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ).
% cSup_upper
thf(fact_1030_cSup__upper2,axiom,
! [X2: nat,X4: set_nat,Y2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ord_less_eq_nat @ Y2 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1031_cSup__upper2,axiom,
! [X2: set_set_nat,X4: set_set_set_nat,Y2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X4 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( condit3670481866171438296et_nat @ X4 )
=> ( ord_le6893508408891458716et_nat @ Y2 @ ( comple548664676211718543et_nat @ X4 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1032_cSup__upper2,axiom,
! [X2: set_nat,X4: set_set_nat,Y2: set_nat] :
( ( member_set_nat @ X2 @ X4 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( condit5477540289124974626et_nat @ X4 )
=> ( ord_less_eq_set_nat @ Y2 @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1033_bdd__above__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( condit5477540289124974626et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( condit5477540289124974626et_nat @ A2 ) ) ) ).
% bdd_above_mono
thf(fact_1034_bdd__above__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( condit2214826472909112428ve_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( condit2214826472909112428ve_nat @ A2 ) ) ) ).
% bdd_above_mono
thf(fact_1035_bdd__above_Ounfold,axiom,
( condit3670481866171438296et_nat
= ( ^ [A3: set_set_set_nat] :
? [M2: set_set_nat] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A3 )
=> ( ord_le6893508408891458716et_nat @ X3 @ M2 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1036_bdd__above_Ounfold,axiom,
( condit5477540289124974626et_nat
= ( ^ [A3: set_set_nat] :
? [M2: set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( ord_less_eq_set_nat @ X3 @ M2 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1037_bdd__above_OE,axiom,
! [A2: set_nat] :
( ( condit2214826472909112428ve_nat @ A2 )
=> ~ ! [M3: nat] :
~ ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_1038_bdd__above_OE,axiom,
! [A2: set_set_set_nat] :
( ( condit3670481866171438296et_nat @ A2 )
=> ~ ! [M3: set_set_nat] :
~ ! [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X6 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_1039_bdd__above_OE,axiom,
! [A2: set_set_nat] :
( ( condit5477540289124974626et_nat @ A2 )
=> ~ ! [M3: set_nat] :
~ ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A2 )
=> ( ord_less_eq_set_nat @ X6 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_1040_finite__subset__Union,axiom,
! [A2: set_set_nat,B8: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ ( comple548664676211718543et_nat @ B8 ) )
=> ~ ! [F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ( ord_le9131159989063066194et_nat @ F3 @ B8 )
=> ~ ( ord_le6893508408891458716et_nat @ A2 @ ( comple548664676211718543et_nat @ F3 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1041_finite__subset__Union,axiom,
! [A2: set_nat,B8: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B8 ) )
=> ~ ! [F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( ord_le6893508408891458716et_nat @ F3 @ B8 )
=> ~ ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ F3 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1042_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1043_finite__has__maximal,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1044_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1045_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1046_finite__has__minimal,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1047_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1048_le__cSup__finite,axiom,
! [X4: set_nat,X2: nat] :
( ( finite_finite_nat @ X4 )
=> ( ( member_nat @ X2 @ X4 )
=> ( ord_less_eq_nat @ X2 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ).
% le_cSup_finite
thf(fact_1049_le__cSup__finite,axiom,
! [X4: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ X4 )
=> ( ( member_set_set_nat @ X2 @ X4 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( comple548664676211718543et_nat @ X4 ) ) ) ) ).
% le_cSup_finite
thf(fact_1050_le__cSup__finite,axiom,
! [X4: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ X4 )
=> ( ( member_set_nat @ X2 @ X4 )
=> ( ord_less_eq_set_nat @ X2 @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ).
% le_cSup_finite
thf(fact_1051_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A6: nat] :
( A
= ( insert_nat @ A6 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1052_finite_Ocases,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ~ ! [A8: set_set_nat] :
( ? [A6: set_nat] :
( A
= ( insert_set_nat @ A6 @ A8 ) )
=> ~ ( finite1152437895449049373et_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1053_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A3: set_nat,B5: nat] :
( ( A4
= ( insert_nat @ B5 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1054_finite_Osimps,axiom,
( finite1152437895449049373et_nat
= ( ^ [A4: set_set_nat] :
( ( A4 = bot_bot_set_set_nat )
| ? [A3: set_set_nat,B5: set_nat] :
( ( A4
= ( insert_set_nat @ B5 @ A3 ) )
& ( finite1152437895449049373et_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1055_finite__induct,axiom,
! [F2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ X @ F4 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1056_finite__induct,axiom,
! [F2: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [X: set_nat,F4: set_set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ~ ( member_set_nat @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_set_nat @ X @ F4 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1057_finite__ne__induct,axiom,
! [F2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X: nat] : ( P2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ! [X: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ X @ F4 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1058_finite__ne__induct,axiom,
! [F2: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( F2 != bot_bot_set_set_nat )
=> ( ! [X: set_nat] : ( P2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
=> ( ! [X: set_nat,F4: set_set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( F4 != bot_bot_set_set_nat )
=> ( ~ ( member_set_nat @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_set_nat @ X @ F4 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1059_infinite__finite__induct,axiom,
! [P2: set_nat > $o,A2: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P2 @ A8 ) )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ X @ F4 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1060_infinite__finite__induct,axiom,
! [P2: set_set_nat > $o,A2: set_set_nat] :
( ! [A8: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A8 )
=> ( P2 @ A8 ) )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [X: set_nat,F4: set_set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ~ ( member_set_nat @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_set_nat @ X @ F4 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1061_cSup__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( B2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A2 )
=> ( ! [B7: nat] :
( ( member_nat @ B7 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_less_eq_nat @ B7 @ X6 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B2 ) @ ( complete_Sup_Sup_nat @ A2 ) ) ) ) ) ).
% cSup_mono
thf(fact_1062_cSup__mono,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( B2 != bot_bo7198184520161983622et_nat )
=> ( ( condit3670481866171438296et_nat @ A2 )
=> ( ! [B7: set_set_nat] :
( ( member_set_set_nat @ B7 @ B2 )
=> ? [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ A2 )
& ( ord_le6893508408891458716et_nat @ B7 @ X6 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ B2 ) @ ( comple548664676211718543et_nat @ A2 ) ) ) ) ) ).
% cSup_mono
thf(fact_1063_cSup__mono,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( B2 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ A2 )
=> ( ! [B7: set_nat] :
( ( member_set_nat @ B7 @ B2 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A2 )
& ( ord_less_eq_set_nat @ B7 @ X6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ B2 ) @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ) ).
% cSup_mono
thf(fact_1064_cSup__le__iff,axiom,
! [S: set_nat,A: nat] :
( ( S != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S ) @ A )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ( ord_less_eq_nat @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1065_cSup__le__iff,axiom,
! [S: set_set_set_nat,A: set_set_nat] :
( ( S != bot_bo7198184520161983622et_nat )
=> ( ( condit3670481866171438296et_nat @ S )
=> ( ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ S ) @ A )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ S )
=> ( ord_le6893508408891458716et_nat @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1066_cSup__le__iff,axiom,
! [S: set_set_nat,A: set_nat] :
( ( S != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ S )
=> ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ S ) @ A )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ S )
=> ( ord_less_eq_set_nat @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1067_less__cSup__iff,axiom,
! [X4: set_nat,Y2: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ( ord_less_nat @ Y2 @ ( complete_Sup_Sup_nat @ X4 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ X4 )
& ( ord_less_nat @ Y2 @ X3 ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_1068_finite__Sup__less__iff,axiom,
! [X4: set_nat,A: nat] :
( ( finite_finite_nat @ X4 )
=> ( ( X4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X4 ) @ A )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ( ord_less_nat @ X3 @ A ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_1069_finite__subset__induct_H,axiom,
! [F2: set_set_nat,A2: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( ord_le6893508408891458716et_nat @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [A6: set_nat,F4: set_set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( member_set_nat @ A6 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ F4 @ A2 )
=> ( ~ ( member_set_nat @ A6 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_set_nat @ A6 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1070_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A6: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ( ord_less_eq_set_nat @ F4 @ A2 )
=> ( ~ ( member_nat @ A6 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ A6 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1071_finite__subset__induct,axiom,
! [F2: set_set_nat,A2: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( ord_le6893508408891458716et_nat @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [A6: set_nat,F4: set_set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( member_set_nat @ A6 @ A2 )
=> ( ~ ( member_set_nat @ A6 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_set_nat @ A6 @ F4 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1072_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A6: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ~ ( member_nat @ A6 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ A6 @ F4 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1073_finite__Sup__in,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat,Y: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( ( member_set_set_nat @ Y @ A2 )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y ) @ A2 ) ) )
=> ( member_set_set_nat @ ( comple548664676211718543et_nat @ A2 ) @ A2 ) ) ) ) ).
% finite_Sup_in
thf(fact_1074_finite__Sup__in,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [X: set_nat,Y: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( member_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ A2 ) ) )
=> ( member_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ A2 ) ) ) ) ).
% finite_Sup_in
thf(fact_1075_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1076_infinite__remove,axiom,
! [S: set_set_nat,A: set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ S @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1077_infinite__coinduct,axiom,
! [X4: set_nat > $o,A2: set_nat] :
( ( X4 @ A2 )
=> ( ! [A8: set_nat] :
( ( X4 @ A8 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A8 )
& ( ( X4 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1078_infinite__coinduct,axiom,
! [X4: set_set_nat > $o,A2: set_set_nat] :
( ( X4 @ A2 )
=> ( ! [A8: set_set_nat] :
( ( X4 @ A8 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A8 )
& ( ( X4 @ ( minus_2163939370556025621et_nat @ A8 @ ( insert_set_nat @ X6 @ bot_bot_set_set_nat ) ) )
| ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A8 @ ( insert_set_nat @ X6 @ bot_bot_set_set_nat ) ) ) ) ) )
=> ~ ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1079_finite__empty__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A6: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A6 @ A8 )
=> ( ( P2 @ A8 )
=> ( P2 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1080_finite__empty__induct,axiom,
! [A2: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A6: set_nat,A8: set_set_nat] :
( ( finite1152437895449049373et_nat @ A8 )
=> ( ( member_set_nat @ A6 @ A8 )
=> ( ( P2 @ A8 )
=> ( P2 @ ( minus_2163939370556025621et_nat @ A8 @ ( insert_set_nat @ A6 @ bot_bot_set_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1081_ring__of__sets_Ofinite__Union,axiom,
! [Omega2: set_nat,M: set_set_nat,X4: set_set_nat] :
( ( sigma_8325262026724180568ts_nat @ Omega2 @ M )
=> ( ( finite1152437895449049373et_nat @ X4 )
=> ( ( ord_le6893508408891458716et_nat @ X4 @ M )
=> ( member_set_nat @ ( comple7399068483239264473et_nat @ X4 ) @ M ) ) ) ) ).
% ring_of_sets.finite_Union
thf(fact_1082_cSup__subset__mono,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ( condit3670481866171438296et_nat @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1083_cSup__subset__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1084_cSup__subset__mono,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A2 ) @ ( complete_Sup_Sup_nat @ B2 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1085_cSup__insert,axiom,
! [X4: set_set_set_nat,A: set_set_nat] :
( ( X4 != bot_bo7198184520161983622et_nat )
=> ( ( condit3670481866171438296et_nat @ X4 )
=> ( ( comple548664676211718543et_nat @ ( insert_set_set_nat @ A @ X4 ) )
= ( sup_sup_set_set_nat @ A @ ( comple548664676211718543et_nat @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_1086_cSup__insert,axiom,
! [X4: set_nat,A: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X4 ) )
= ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_1087_cSup__insert,axiom,
! [X4: set_set_nat,A: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ X4 )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X4 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_1088_cSup__insert__If,axiom,
! [X4: set_set_set_nat,A: set_set_nat] :
( ( condit3670481866171438296et_nat @ X4 )
=> ( ( ( X4 = bot_bo7198184520161983622et_nat )
=> ( ( comple548664676211718543et_nat @ ( insert_set_set_nat @ A @ X4 ) )
= A ) )
& ( ( X4 != bot_bo7198184520161983622et_nat )
=> ( ( comple548664676211718543et_nat @ ( insert_set_set_nat @ A @ X4 ) )
= ( sup_sup_set_set_nat @ A @ ( comple548664676211718543et_nat @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_1089_cSup__insert__If,axiom,
! [X4: set_nat,A: nat] :
( ( condit2214826472909112428ve_nat @ X4 )
=> ( ( ( X4 = bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X4 ) )
= A ) )
& ( ( X4 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X4 ) )
= ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_1090_cSup__insert__If,axiom,
! [X4: set_set_nat,A: set_nat] :
( ( condit5477540289124974626et_nat @ X4 )
=> ( ( ( X4 = bot_bot_set_set_nat )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X4 ) )
= A ) )
& ( ( X4 != bot_bot_set_set_nat )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X4 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_1091_cSup__union__distrib,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ( condit3670481866171438296et_nat @ A2 )
=> ( ( B2 != bot_bo7198184520161983622et_nat )
=> ( ( condit3670481866171438296et_nat @ B2 )
=> ( ( comple548664676211718543et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ ( comple548664676211718543et_nat @ A2 ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1092_cSup__union__distrib,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B2 )
=> ( ( complete_Sup_Sup_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A2 ) @ ( complete_Sup_Sup_nat @ B2 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1093_cSup__union__distrib,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ A2 )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ B2 )
=> ( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1094_finite__remove__induct,axiom,
! [B2: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [A8: set_set_nat] :
( ( finite1152437895449049373et_nat @ A8 )
=> ( ( A8 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ A8 @ B2 )
=> ( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A8 )
=> ( P2 @ ( minus_2163939370556025621et_nat @ A8 @ ( insert_set_nat @ X6 @ bot_bot_set_set_nat ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1095_finite__remove__induct,axiom,
! [B2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A8 )
=> ( P2 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1096_remove__induct,axiom,
! [P2: set_set_nat > $o,B2: set_set_nat] :
( ( P2 @ bot_bot_set_set_nat )
=> ( ( ~ ( finite1152437895449049373et_nat @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A8: set_set_nat] :
( ( finite1152437895449049373et_nat @ A8 )
=> ( ( A8 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ A8 @ B2 )
=> ( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A8 )
=> ( P2 @ ( minus_2163939370556025621et_nat @ A8 @ ( insert_set_nat @ X6 @ bot_bot_set_set_nat ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1097_remove__induct,axiom,
! [P2: set_nat > $o,B2: set_nat] :
( ( P2 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A8 )
=> ( P2 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1098_finite__induct__select,axiom,
! [S: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S )
=> ( ( P2 @ T3 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ ( minus_minus_set_nat @ S @ T3 ) )
& ( P2 @ ( insert_nat @ X6 @ T3 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1099_finite__induct__select,axiom,
! [S: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( P2 @ bot_bot_set_set_nat )
=> ( ! [T3: set_set_nat] :
( ( ord_less_set_set_nat @ T3 @ S )
=> ( ( P2 @ T3 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( minus_2163939370556025621et_nat @ S @ T3 ) )
& ( P2 @ ( insert_set_nat @ X6 @ T3 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1100_le__cSup__iff,axiom,
! [A2: set_nat,X2: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ ( complete_Sup_Sup_nat @ A2 ) )
= ( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_nat @ Y4 @ X3 ) ) ) ) ) ) ) ).
% le_cSup_iff
thf(fact_1101_finite__linorder__max__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [B7: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A8 )
=> ( ord_less_nat @ X6 @ B7 ) )
=> ( ( P2 @ A8 )
=> ( P2 @ ( insert_nat @ B7 @ A8 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1102_finite__linorder__min__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [B7: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A8 )
=> ( ord_less_nat @ B7 @ X6 ) )
=> ( ( P2 @ A8 )
=> ( P2 @ ( insert_nat @ B7 @ A8 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1103_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1104_ex__min__if__finite,axiom,
! [S: set_set_nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( S != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ S )
& ~ ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ S )
& ( ord_less_set_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1105_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X4 )
& ( ord_less_nat @ X @ Xa ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_1106_finite__transitivity__chain,axiom,
! [A2: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X: nat] :
~ ( R2 @ X @ X )
=> ( ! [X: nat,Y: nat,Z3: nat] :
( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z3 )
=> ( R2 @ X @ Z3 ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
& ( R2 @ X @ Y6 ) ) )
=> ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1107_finite__transitivity__chain,axiom,
! [A2: set_set_nat,R2: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [X: set_nat] :
~ ( R2 @ X @ X )
=> ( ! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z3 )
=> ( R2 @ X @ Z3 ) ) )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ A2 )
& ( R2 @ X @ Y6 ) ) )
=> ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1108_Sup__fin_Oinsert__remove,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) )
= bot_bo7198184520161983622et_nat )
=> ( ( lattic7928989940735914181et_nat @ ( insert_set_set_nat @ X2 @ A2 ) )
= X2 ) )
& ( ( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) )
!= bot_bo7198184520161983622et_nat )
=> ( ( lattic7928989940735914181et_nat @ ( insert_set_set_nat @ X2 @ A2 ) )
= ( sup_sup_set_set_nat @ X2 @ ( lattic7928989940735914181et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1109_Sup__fin_Oinsert__remove,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A2 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A2 ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1110_Sup__fin_Oinsert__remove,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X2 @ A2 ) )
= X2 ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X2 @ A2 ) )
= ( sup_sup_set_nat @ X2 @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1111_Sup__fin_Oremove,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ X2 @ A2 )
=> ( ( ( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) )
= bot_bo7198184520161983622et_nat )
=> ( ( lattic7928989940735914181et_nat @ A2 )
= X2 ) )
& ( ( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) )
!= bot_bo7198184520161983622et_nat )
=> ( ( lattic7928989940735914181et_nat @ A2 )
= ( sup_sup_set_set_nat @ X2 @ ( lattic7928989940735914181et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1112_Sup__fin_Oremove,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1113_Sup__fin_Oremove,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ A2 )
= X2 ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ A2 )
= ( sup_sup_set_nat @ X2 @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1114_Sup__fin_Osingleton,axiom,
! [X2: nat] :
( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Sup_fin.singleton
thf(fact_1115_Sup__fin_Osingleton,axiom,
! [X2: set_nat] :
( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= X2 ) ).
% Sup_fin.singleton
thf(fact_1116_Sup__fin_Oinsert,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ( lattic7928989940735914181et_nat @ ( insert_set_set_nat @ X2 @ A2 ) )
= ( sup_sup_set_set_nat @ X2 @ ( lattic7928989940735914181et_nat @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1117_Sup__fin_Oinsert,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A2 ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1118_Sup__fin_Oinsert,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X2 @ A2 ) )
= ( sup_sup_set_nat @ X2 @ ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1119_Sup__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1120_Sup__fin_OcoboundedI,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ( ord_le6893508408891458716et_nat @ A @ ( lattic7928989940735914181et_nat @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1121_Sup__fin_OcoboundedI,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ A @ ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1122_Sup__fin_Oin__idem,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1123_Sup__fin_Oin__idem,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ X2 @ A2 )
=> ( ( sup_sup_set_set_nat @ X2 @ ( lattic7928989940735914181et_nat @ A2 ) )
= ( lattic7928989940735914181et_nat @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1124_Sup__fin_Oin__idem,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( ( sup_sup_set_nat @ X2 @ ( lattic3835124923745554447et_nat @ A2 ) )
= ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1125_Sup__fin_Obounded__iff,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1126_Sup__fin_Obounded__iff,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ( ord_le6893508408891458716et_nat @ ( lattic7928989940735914181et_nat @ A2 ) @ X2 )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1127_Sup__fin_Obounded__iff,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ X2 )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1128_Sup__fin_OboundedI,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ord_less_eq_nat @ A6 @ X2 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1129_Sup__fin_OboundedI,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ! [A6: set_set_nat] :
( ( member_set_set_nat @ A6 @ A2 )
=> ( ord_le6893508408891458716et_nat @ A6 @ X2 ) )
=> ( ord_le6893508408891458716et_nat @ ( lattic7928989940735914181et_nat @ A2 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1130_Sup__fin_OboundedI,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A2 )
=> ( ord_less_eq_set_nat @ A6 @ X2 ) )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1131_Sup__fin_OboundedE,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X2 )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A2 )
=> ( ord_less_eq_nat @ A9 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1132_Sup__fin_OboundedE,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ( ord_le6893508408891458716et_nat @ ( lattic7928989940735914181et_nat @ A2 ) @ X2 )
=> ! [A9: set_set_nat] :
( ( member_set_set_nat @ A9 @ A2 )
=> ( ord_le6893508408891458716et_nat @ A9 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1133_Sup__fin_OboundedE,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ X2 )
=> ! [A9: set_nat] :
( ( member_set_nat @ A9 @ A2 )
=> ( ord_less_eq_set_nat @ A9 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1134_Sup__fin__Sup,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ A2 )
= ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% Sup_fin_Sup
thf(fact_1135_cSup__eq__Sup__fin,axiom,
! [X4: set_nat] :
( ( finite_finite_nat @ X4 )
=> ( ( X4 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ X4 )
= ( lattic1093996805478795353in_nat @ X4 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_1136_cSup__eq__Sup__fin,axiom,
! [X4: set_set_nat] :
( ( finite1152437895449049373et_nat @ X4 )
=> ( ( X4 != bot_bot_set_set_nat )
=> ( ( comple7399068483239264473et_nat @ X4 )
= ( lattic3835124923745554447et_nat @ X4 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_1137_Sup__fin_Osubset__imp,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( lattic7928989940735914181et_nat @ A2 ) @ ( lattic7928989940735914181et_nat @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1138_Sup__fin_Osubset__imp,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ ( lattic3835124923745554447et_nat @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1139_Sup__fin_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1140_Sup__fin_Osubset,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( B2 != bot_bo7198184520161983622et_nat )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( sup_sup_set_set_nat @ ( lattic7928989940735914181et_nat @ B2 ) @ ( lattic7928989940735914181et_nat @ A2 ) )
= ( lattic7928989940735914181et_nat @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1141_Sup__fin_Osubset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ B2 ) @ ( lattic3835124923745554447et_nat @ A2 ) )
= ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1142_Sup__fin_Osubset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B2 ) @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1143_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ( lattic7928989940735914181et_nat @ ( insert_set_set_nat @ X2 @ A2 ) )
= ( sup_sup_set_set_nat @ X2 @ ( lattic7928989940735914181et_nat @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1144_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X2 @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A2 ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1145_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X2 @ A2 ) )
= ( sup_sup_set_nat @ X2 @ ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1146_Sup__fin_Oclosed,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat,Y: set_set_nat] : ( member_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y ) @ ( insert_set_set_nat @ X @ ( insert_set_set_nat @ Y @ bot_bo7198184520161983622et_nat ) ) )
=> ( member_set_set_nat @ ( lattic7928989940735914181et_nat @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1147_Sup__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X: nat,Y: nat] : ( member_nat @ ( sup_sup_nat @ X @ Y ) @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1148_Sup__fin_Oclosed,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [X: set_nat,Y: set_nat] : ( member_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( insert_set_nat @ X @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) )
=> ( member_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1149_Sup__fin_Ounion,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ( B2 != bot_bo7198184520161983622et_nat )
=> ( ( lattic7928989940735914181et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ ( lattic7928989940735914181et_nat @ A2 ) @ ( lattic7928989940735914181et_nat @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1150_Sup__fin_Ounion,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1151_Sup__fin_Ounion,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ ( lattic3835124923745554447et_nat @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1152_Inf__fin_Oremove,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1153_Inf__fin_Oremove,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ A2 )
= X2 ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ A2 )
= ( inf_inf_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1154_Inf__fin_Oinsert__remove,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A2 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A2 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1155_Inf__fin_Oinsert__remove,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X2 @ A2 ) )
= X2 ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X2 @ A2 ) )
= ( inf_inf_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1156_Inf__fin_Oinsert,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A2 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1157_Inf__fin_Oinsert,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X2 @ A2 ) )
= ( inf_inf_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1158_Inf__fin_Osingleton,axiom,
! [X2: nat] :
( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Inf_fin.singleton
thf(fact_1159_Inf__fin_Osingleton,axiom,
! [X2: set_nat] :
( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= X2 ) ).
% Inf_fin.singleton
thf(fact_1160_sup__Inf__absorb,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1161_sup__Inf__absorb,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ( ( sup_sup_set_set_nat @ ( lattic700688560247204575et_nat @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1162_sup__Inf__absorb,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( ( sup_sup_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1163_Inf__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1164_Inf__fin_OcoboundedI,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( lattic700688560247204575et_nat @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1165_Inf__fin_OcoboundedI,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1166_Inf__fin_OboundedE,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A2 ) )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A2 )
=> ( ord_less_eq_nat @ X2 @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1167_Inf__fin_OboundedE,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ ( lattic700688560247204575et_nat @ A2 ) )
=> ! [A9: set_set_nat] :
( ( member_set_set_nat @ A9 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1168_Inf__fin_OboundedE,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ A2 ) )
=> ! [A9: set_nat] :
( ( member_set_nat @ A9 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1169_Inf__fin_OboundedI,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ord_less_eq_nat @ X2 @ A6 ) )
=> ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1170_Inf__fin_OboundedI,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ! [A6: set_set_nat] :
( ( member_set_set_nat @ A6 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ A6 ) )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( lattic700688560247204575et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1171_Inf__fin_OboundedI,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ A6 ) )
=> ( ord_less_eq_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1172_Inf__fin_Obounded__iff,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1173_Inf__fin_Obounded__iff,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ ( lattic700688560247204575et_nat @ A2 ) )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1174_Inf__fin_Obounded__iff,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ A2 ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1175_Inf__fin_Osubset__imp,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ( finite6739761609112101331et_nat @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( lattic700688560247204575et_nat @ B2 ) @ ( lattic700688560247204575et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1176_Inf__fin_Osubset__imp,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B2 ) @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1177_Inf__fin_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B2 ) @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1178_Inf__fin_Osubset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ B2 ) @ ( lattic3014633134055518761et_nat @ A2 ) )
= ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1179_Inf__fin_Osubset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B2 ) @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1180_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X2 @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A2 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1181_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X2 @ A2 ) )
= ( inf_inf_set_nat @ X2 @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1182_Inf__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X: nat,Y: nat] : ( member_nat @ ( inf_inf_nat @ X @ Y ) @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1183_Inf__fin_Oclosed,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [X: set_nat,Y: set_nat] : ( member_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( insert_set_nat @ X @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) )
=> ( member_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1184_Inf__fin_Ounion,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic5238388535129920115in_nat @ B2 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1185_Inf__fin_Ounion,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ ( lattic3014633134055518761et_nat @ B2 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1186_Inf__fin__le__Sup__fin,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1187_Inf__fin__le__Sup__fin,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ ( lattic700688560247204575et_nat @ A2 ) @ ( lattic7928989940735914181et_nat @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1188_Inf__fin__le__Sup__fin,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1189_cSUP__union,axiom,
! [A2: set_nat,F: nat > set_set_nat,B2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit3670481866171438296et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( condit3670481866171438296et_nat @ ( image_2194112158459175443et_nat @ F @ B2 ) )
=> ( ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ ( sup_sup_set_nat @ A2 @ B2 ) ) )
= ( sup_sup_set_set_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1190_cSUP__union,axiom,
! [A2: set_nat,F: nat > set_nat,B2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A2 ) )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ B2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A2 @ B2 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1191_cSUP__union,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat,B2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ( condit3670481866171438296et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( condit3670481866171438296et_nat @ ( image_6725021117256019401et_nat @ F @ B2 ) )
=> ( ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ F @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) )
= ( sup_sup_set_set_nat @ ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) ) @ ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1192_cSUP__union,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,B2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_7916887816326733075et_nat @ F @ B2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1193_cSUP__insert,axiom,
! [A2: set_nat,F: nat > set_set_nat,A: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit3670481866171438296et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ ( insert_nat @ A @ A2 ) ) )
= ( sup_sup_set_set_nat @ ( F @ A ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_1194_cSUP__insert,axiom,
! [A2: set_nat,F: nat > set_nat,A: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( insert_nat @ A @ A2 ) ) )
= ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_1195_cSUP__insert,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat,A: set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ( condit3670481866171438296et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) )
=> ( ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ F @ ( insert_set_nat @ A @ A2 ) ) )
= ( sup_sup_set_set_nat @ ( F @ A ) @ ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_1196_cSUP__insert,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,A: set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ ( insert_set_nat @ A @ A2 ) ) )
= ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_1197_cSUP__subset__mono,axiom,
! [A2: set_set_nat,G2: set_nat > set_set_nat,B2: set_set_nat,F: set_nat > set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ( condit3670481866171438296et_nat @ ( image_6725021117256019401et_nat @ G2 @ B2 ) )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) ) @ ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ G2 @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1198_cSUP__subset__mono,axiom,
! [A2: set_set_nat,G2: set_nat > set_nat,B2: set_set_nat,F: set_nat > set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_7916887816326733075et_nat @ G2 @ B2 ) )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G2 @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1199_cSUP__subset__mono,axiom,
! [A2: set_nat,G2: nat > set_set_nat,B2: set_nat,F: nat > set_set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit3670481866171438296et_nat @ ( image_2194112158459175443et_nat @ G2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ G2 @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1200_cSUP__subset__mono,axiom,
! [A2: set_nat,G2: nat > set_nat,B2: set_nat,F: nat > set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1201_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_1202_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_1203_image__eqI,axiom,
! [B: nat,F: set_nat > nat,X2: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1204_image__eqI,axiom,
! [B: set_nat,F: set_nat > set_nat,X2: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1205_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1206_image__is__empty,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( ( image_set_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_1207_image__is__empty,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( ( image_nat_set_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1208_image__is__empty,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_1209_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1210_empty__is__image,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( bot_bot_set_nat
= ( image_set_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_1211_empty__is__image,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1212_empty__is__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_1213_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1214_image__empty,axiom,
! [F: nat > set_nat] :
( ( image_nat_set_nat @ F @ bot_bot_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_1215_image__empty,axiom,
! [F: set_nat > nat] :
( ( image_set_nat_nat @ F @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1216_image__empty,axiom,
! [F: set_nat > set_nat] :
( ( image_7916887816326733075et_nat @ F @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_1217_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1218_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A2 ) )
= ( image_nat_set_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1219_insert__image,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_set_nat_nat @ F @ A2 ) )
= ( image_set_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1220_insert__image,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1221_image__insert,axiom,
! [F: nat > nat,A: nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1222_image__insert,axiom,
! [F: nat > set_nat,A: nat,B2: set_nat] :
( ( image_nat_set_nat @ F @ ( insert_nat @ A @ B2 ) )
= ( insert_set_nat @ ( F @ A ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1223_image__insert,axiom,
! [F: set_nat > nat,A: set_nat,B2: set_set_nat] :
( ( image_set_nat_nat @ F @ ( insert_set_nat @ A @ B2 ) )
= ( insert_nat @ ( F @ A ) @ ( image_set_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1224_image__insert,axiom,
! [F: set_nat > set_nat,A: set_nat,B2: set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( insert_set_nat @ A @ B2 ) )
= ( insert_set_nat @ ( F @ A ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1225_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S3: set_nat_o,X3: nat] : ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1226_Sup__SUP__eq,axiom,
( comple3806919086088850358_nat_o
= ( ^ [S3: set_set_nat_o,X3: set_nat] : ( member_set_nat @ X3 @ ( comple548664676211718543et_nat @ ( image_4687162037615663680et_nat @ collect_set_nat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1227_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_1228_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_1229_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B: nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1230_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1231_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_1232_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_1233_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_set_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1234_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1235_subset__image__iff,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_7916887816326733075et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1236_subset__image__iff,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1237_subset__image__iff,axiom,
! [B2: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_set_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1238_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_1239_subset__imageE,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ~ ! [C4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C4 @ A2 )
=> ( B2
!= ( image_7916887816326733075et_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1240_subset__imageE,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B2
!= ( image_nat_set_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1241_subset__imageE,axiom,
! [B2: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
=> ~ ! [C4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C4 @ A2 )
=> ( B2
!= ( image_set_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1242_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1243_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B2: set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1244_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,B2: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1245_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_1246_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > nat,B2: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1247_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1248_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ ( image_set_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1249_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1250_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_1251_image__Un,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1252_image__Un,axiom,
! [F: set_nat > nat,A2: set_set_nat,B2: set_set_nat] :
( ( image_set_nat_nat @ F @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ ( image_set_nat_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1253_image__Un,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_nat] :
( ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1254_image__Un,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1255_all__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P2: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( P2 @ ( image_7916887816326733075et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1256_all__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P2: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P2 @ ( image_nat_set_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1257_all__subset__image,axiom,
! [F: set_nat > nat,A2: set_set_nat,P2: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_set_nat_nat @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( P2 @ ( image_set_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1258_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P2: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P2 @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1259_bdd__above_OI2,axiom,
! [A2: set_nat,F: nat > set_set_nat,M: set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ M ) )
=> ( condit3670481866171438296et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ).
% bdd_above.I2
thf(fact_1260_bdd__above_OI2,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat,M: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ M ) )
=> ( condit3670481866171438296et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ).
% bdd_above.I2
thf(fact_1261_bdd__above_OI2,axiom,
! [A2: set_nat,F: nat > set_nat,M: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ M ) )
=> ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% bdd_above.I2
thf(fact_1262_bdd__above_OI2,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,M: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ M ) )
=> ( condit5477540289124974626et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ).
% bdd_above.I2
thf(fact_1263_all__finite__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P2: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ ( image_7916887816326733075et_nat @ F @ A2 ) ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A2 ) )
=> ( P2 @ ( image_7916887816326733075et_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1264_all__finite__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P2: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) )
=> ( P2 @ ( image_nat_set_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1265_all__finite__subset__image,axiom,
! [F: set_nat > nat,A2: set_set_nat,P2: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_set_nat_nat @ F @ A2 ) ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A2 ) )
=> ( P2 @ ( image_set_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1266_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P2: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) )
=> ( P2 @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1267_ex__finite__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P2: set_set_nat > $o] :
( ( ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
& ( P2 @ B3 ) ) )
= ( ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A2 )
& ( P2 @ ( image_7916887816326733075et_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1268_ex__finite__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P2: set_set_nat > $o] :
( ( ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P2 @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 )
& ( P2 @ ( image_nat_set_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1269_ex__finite__subset__image,axiom,
! [F: set_nat > nat,A2: set_set_nat,P2: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_set_nat_nat @ F @ A2 ) )
& ( P2 @ B3 ) ) )
= ( ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A2 )
& ( P2 @ ( image_set_nat_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1270_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P2: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
& ( P2 @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 )
& ( P2 @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1271_finite__subset__image,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ? [C4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C4 @ A2 )
& ( finite1152437895449049373et_nat @ C4 )
& ( B2
= ( image_7916887816326733075et_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1272_finite__subset__image,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( finite_finite_nat @ C4 )
& ( B2
= ( image_nat_set_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1273_finite__subset__image,axiom,
! [B2: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) )
=> ? [C4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C4 @ A2 )
& ( finite1152437895449049373et_nat @ C4 )
& ( B2
= ( image_set_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1274_finite__subset__image,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( finite_finite_nat @ C4 )
& ( B2
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1275_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G2: nat > set_nat] :
( ! [I: nat] :
( ( member_nat @ I @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I ) @ ( G2 @ X6 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_less_eq_set_nat @ ( G2 @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1276_SUP__eq,axiom,
! [A2: set_nat,B2: set_set_nat,F: nat > set_nat,G2: set_nat > set_nat] :
( ! [I: nat] :
( ( member_nat @ I @ A2 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I ) @ ( G2 @ X6 ) ) ) )
=> ( ! [J: set_nat] :
( ( member_set_nat @ J @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_less_eq_set_nat @ ( G2 @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G2 @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1277_SUP__eq,axiom,
! [A2: set_set_nat,B2: set_nat,F: set_nat > set_nat,G2: nat > set_nat] :
( ! [I: set_nat] :
( ( member_set_nat @ I @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I ) @ ( G2 @ X6 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B2 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A2 )
& ( ord_less_eq_set_nat @ ( G2 @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1278_SUP__eq,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat,G2: set_nat > set_nat] :
( ! [I: set_nat] :
( ( member_set_nat @ I @ A2 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I ) @ ( G2 @ X6 ) ) ) )
=> ( ! [J: set_nat] :
( ( member_set_nat @ J @ B2 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A2 )
& ( ord_less_eq_set_nat @ ( G2 @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G2 @ B2 ) ) ) ) ) ).
% SUP_eq
% Conjectures (1)
thf(conj_0,conjecture,
member_nat @ x @ ( clique5033774636164728513irst_v @ g ) ).
%------------------------------------------------------------------------------