TPTP Problem File: SLH0290^1.p
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- 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 : Median_Method/0000_Median/prob_00057_002136__14627184_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1409 ( 428 unt; 135 typ; 0 def)
% Number of atoms : 4181 (1050 equ; 0 cnn)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 11357 ( 467 ~; 100 |; 273 &;8539 @)
% ( 0 <=>;1978 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 525 ( 525 >; 0 *; 0 +; 0 <<)
% Number of symbols : 127 ( 125 usr; 9 con; 0-4 aty)
% Number of variables : 3786 ( 311 ^;3336 !; 139 ?;3786 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:43:05.972
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Filter__Ofilter_Itf__a_J,type,
filter_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (125)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__a_J,type,
comple6135023378680113637_set_a: set_set_a > set_a ).
thf(sy_c_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_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
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_Itf__a_J,type,
condit3373647341569784514_set_a: set_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001tf__a,type,
condit5209368051240477026bove_a: set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Nat__Onat,type,
condit1738341127787009408ow_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Set__Oset_Itf__a_J,type,
condit8937546108433946286_set_a: set_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001tf__a,type,
condit5901475214736682318elow_a: set_a > $o ).
thf(sy_c_Elementary__Topology_Oclosure_001tf__a,type,
elementary_closure_a: set_a > set_a ).
thf(sy_c_Extended__Real__Limits_Oorder__class_Omono__set_001t__Set__Oset_Itf__a_J,type,
extend347329919781519187_set_a: set_set_a > $o ).
thf(sy_c_Extended__Real__Limits_Oorder__class_Omono__set_001tf__a,type,
extend2808419353335425523_set_a: set_a > $o ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
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_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_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_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_Itf__a_J,type,
semila2496817875450240012_set_a: ( set_a > set_a > set_a ) > set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
sup_sup_a_a_o: ( a > a > $o ) > ( a > a > $o ) > a > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_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_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Nat__Onat,type,
lattic6009151579333465974et_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_Itf__a_J,type,
lattic8986249270076014136_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Median_Ointerval_001t__Nat__Onat,type,
interval_nat: set_nat > $o ).
thf(sy_c_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Median_Oup__ray_001t__Nat__Onat,type,
up_ray_nat: set_nat > $o ).
thf(sy_c_Median_Oup__ray_001tf__a,type,
up_ray_a: set_a > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le8783759336481291722_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_Itf__a_J,type,
ord_less_eq_filter_a: filter_a > filter_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_Itf__a_J,type,
ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > 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_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
set_or6288561110385358355_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001tf__a,type,
set_or672772299803893939Most_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_Itf__a_J,type,
set_or2348907005316661231_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001tf__a,type,
set_or5139330845457685135Than_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Nat__Onat_J,type,
set_or1731685050470061051et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or1796310902737568945et_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or3904034815786525833_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_Itf__a_J,type,
set_or8362275514725411625_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001tf__a,type,
set_ord_atLeast_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001tf__a,type,
set_ord_atMost_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_Itf__a_J,type,
set_or2503527069484367278_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001tf__a,type,
set_or4472690218693186638Most_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_Itf__a_J,type,
set_or6017932776736107018_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001tf__a,type,
set_or5939364468397584554Than_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001tf__a,type,
set_or8632414552788122084Than_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001tf__a,type,
set_ord_lessThan_a: a > set_a ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Nat__Onat,type,
topolo4328251076210115529en_nat: set_nat > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__a,type,
topolo8477419352202985285open_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Nat__Onat,type,
topolo4659099751122792720in_nat: nat > set_nat > filter_nat ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001tf__a,type,
topolo1902352237885396414thin_a: a > set_a > filter_a ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oclosed_001t__Nat__Onat,type,
topolo517082937297594846ed_nat: set_nat > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oclosed_001tf__a,type,
topolo784654279908865136osed_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Ocompact_001tf__a,type,
topolo8439159285038550427pact_a: set_a > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected_001t__Nat__Onat,type,
topolo3071422859925075001ed_nat: set_nat > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected_001tf__a,type,
topolo2370605967727889109cted_a: set_a > $o ).
thf(sy_c_Zorn_Ochain__subset_001tf__a,type,
chain_subset_a: set_set_a > $o ).
thf(sy_c_Zorn_Ochains_001tf__a,type,
chains_a: set_set_a > set_set_set_a ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_Itf__a_J,type,
pred_chain_set_a: set_set_a > ( set_a > set_a > $o ) > set_set_a > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001tf__a,type,
pred_chain_a: set_a > ( a > a > $o ) > set_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (1273)
thf(fact_0_False,axiom,
~ ? [Y: a] :
( ( ord_less_a @ Y @ x )
& ( member_a @ Y @ i ) ) ).
% False
thf(fact_1_b,axiom,
~ ( topolo784654279908865136osed_a @ i ) ).
% b
thf(fact_2__092_060open_062closed_A_123x_O_O_125_092_060close_062,axiom,
topolo784654279908865136osed_a @ ( set_ord_atLeast_a @ x ) ).
% \<open>closed {x..}\<close>
thf(fact_3_assms,axiom,
up_ray_a @ i ).
% assms
thf(fact_4__092_060open_062I_A_061_A_123x_O_O_125_092_060close_062,axiom,
( i
= ( set_ord_atLeast_a @ x ) ) ).
% \<open>I = {x..}\<close>
thf(fact_5_c,axiom,
member_a @ x @ i ).
% c
thf(fact_6__092_060open_062I_A_092_060subseteq_062_A_123x_O_O_125_092_060close_062,axiom,
ord_less_eq_set_a @ i @ ( set_ord_atLeast_a @ x ) ).
% \<open>I \<subseteq> {x..}\<close>
thf(fact_7__092_060open_062_123x_O_O_125_A_092_060subseteq_062_AI_092_060close_062,axiom,
ord_less_eq_set_a @ ( set_ord_atLeast_a @ x ) @ i ).
% \<open>{x..} \<subseteq> I\<close>
thf(fact_8_minf_I7_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ X @ Z )
=> ~ ( ord_less_a @ T @ X ) ) ).
% minf(7)
thf(fact_9_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ~ ( ord_less_nat @ T @ X ) ) ).
% minf(7)
thf(fact_10_minf_I5_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ X @ Z )
=> ( ord_less_a @ X @ T ) ) ).
% minf(5)
thf(fact_11_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_nat @ X @ T ) ) ).
% minf(5)
thf(fact_12_minf_I4_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_13_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_14_minf_I3_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_15_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_16_minf_I2_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z2: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: a] :
! [X: a] :
( ( ord_less_a @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_17_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_18_minf_I1_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z2: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: a] :
! [X2: a] :
( ( ord_less_a @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: a] :
! [X: a] :
( ( ord_less_a @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_19_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_20_pinf_I7_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ Z @ X )
=> ( ord_less_a @ T @ X ) ) ).
% pinf(7)
thf(fact_21_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ord_less_nat @ T @ X ) ) ).
% pinf(7)
thf(fact_22_pinf_I5_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ Z @ X )
=> ~ ( ord_less_a @ X @ T ) ) ).
% pinf(5)
thf(fact_23_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ~ ( ord_less_nat @ X @ T ) ) ).
% pinf(5)
thf(fact_24_up__ray__def,axiom,
( up_ray_nat
= ( ^ [I: set_nat] :
! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ I )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( member_nat @ Y2 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_25_up__ray__def,axiom,
( up_ray_a
= ( ^ [I: set_a] :
! [X3: a,Y2: a] :
( ( member_a @ X3 @ I )
=> ( ( ord_less_eq_a @ X3 @ Y2 )
=> ( member_a @ Y2 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_26_pinf_I6_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ Z @ X )
=> ~ ( ord_less_eq_a @ X @ T ) ) ).
% pinf(6)
thf(fact_27_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ T ) ) ).
% pinf(6)
thf(fact_28_pinf_I8_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ Z @ X )
=> ( ord_less_eq_a @ T @ X ) ) ).
% pinf(8)
thf(fact_29_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ord_less_eq_nat @ T @ X ) ) ).
% pinf(8)
thf(fact_30_minf_I6_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ X @ Z )
=> ( ord_less_eq_a @ X @ T ) ) ).
% minf(6)
thf(fact_31_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ T ) ) ).
% minf(6)
thf(fact_32_minf_I8_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ X @ Z )
=> ~ ( ord_less_eq_a @ T @ X ) ) ).
% minf(8)
thf(fact_33_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ T @ X ) ) ).
% minf(8)
thf(fact_34_pinf_I1_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z2: a] :
! [X2: a] :
( ( ord_less_a @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: a] :
! [X2: a] :
( ( ord_less_a @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: a] :
! [X: a] :
( ( ord_less_a @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_35_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_36_pinf_I2_J,axiom,
! [P: a > $o,P2: a > $o,Q: a > $o,Q2: a > $o] :
( ? [Z2: a] :
! [X2: a] :
( ( ord_less_a @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: a] :
! [X2: a] :
( ( ord_less_a @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: a] :
! [X: a] :
( ( ord_less_a @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_37_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_38_pinf_I3_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_39_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_40_pinf_I4_J,axiom,
! [T: a] :
? [Z: a] :
! [X: a] :
( ( ord_less_a @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_41_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X: nat] :
( ( ord_less_nat @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_42_atLeast__subset__iff,axiom,
! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or1796310902737568945et_nat @ X4 ) @ ( set_or1796310902737568945et_nat @ Y3 ) )
= ( ord_le6893508408891458716et_nat @ Y3 @ X4 ) ) ).
% atLeast_subset_iff
thf(fact_43_atLeast__subset__iff,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or3904034815786525833_set_a @ X4 ) @ ( set_or3904034815786525833_set_a @ Y3 ) )
= ( ord_le3724670747650509150_set_a @ Y3 @ X4 ) ) ).
% atLeast_subset_iff
thf(fact_44_atLeast__subset__iff,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or1731685050470061051et_nat @ X4 ) @ ( set_or1731685050470061051et_nat @ Y3 ) )
= ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ).
% atLeast_subset_iff
thf(fact_45_atLeast__subset__iff,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ X4 ) @ ( set_ord_atLeast_nat @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% atLeast_subset_iff
thf(fact_46_atLeast__subset__iff,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or8362275514725411625_set_a @ X4 ) @ ( set_or8362275514725411625_set_a @ Y3 ) )
= ( ord_less_eq_set_a @ Y3 @ X4 ) ) ).
% atLeast_subset_iff
thf(fact_47_atLeast__subset__iff,axiom,
! [X4: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ X4 ) @ ( set_ord_atLeast_a @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X4 ) ) ).
% atLeast_subset_iff
thf(fact_48_atLeast__iff,axiom,
! [I2: set_set_nat,K: set_set_nat] :
( ( member_set_set_nat @ I2 @ ( set_or1796310902737568945et_nat @ K ) )
= ( ord_le6893508408891458716et_nat @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_49_atLeast__iff,axiom,
! [I2: set_set_a,K: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or3904034815786525833_set_a @ K ) )
= ( ord_le3724670747650509150_set_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_50_atLeast__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or1731685050470061051et_nat @ K ) )
= ( ord_less_eq_set_nat @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_51_atLeast__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atLeast_nat @ K ) )
= ( ord_less_eq_nat @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_52_atLeast__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_or8362275514725411625_set_a @ K ) )
= ( ord_less_eq_set_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_53_atLeast__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_atLeast_a @ K ) )
= ( ord_less_eq_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_54_atLeast__eq__iff,axiom,
! [X4: set_a,Y3: set_a] :
( ( ( set_or8362275514725411625_set_a @ X4 )
= ( set_or8362275514725411625_set_a @ Y3 ) )
= ( X4 = Y3 ) ) ).
% atLeast_eq_iff
thf(fact_55_atLeast__eq__iff,axiom,
! [X4: nat,Y3: nat] :
( ( ( set_ord_atLeast_nat @ X4 )
= ( set_ord_atLeast_nat @ Y3 ) )
= ( X4 = Y3 ) ) ).
% atLeast_eq_iff
thf(fact_56_atLeast__eq__iff,axiom,
! [X4: a,Y3: a] :
( ( ( set_ord_atLeast_a @ X4 )
= ( set_ord_atLeast_a @ Y3 ) )
= ( X4 = Y3 ) ) ).
% atLeast_eq_iff
thf(fact_57_psubsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_58_psubsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_59_psubsetI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_60_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_61_subset__antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_62_subset__antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_63_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_64_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_65_subsetI,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( member_set_set_a @ X2 @ B ) )
=> ( ord_le5722252365846178494_set_a @ A @ B ) ) ).
% subsetI
thf(fact_66_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ X2 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_67_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_68_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_69_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_70_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_71_dual__order_Orefl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_72_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_73_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_74_dual__order_Orefl,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_75_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_76_order__refl,axiom,
! [X4: set_set_nat] : ( ord_le6893508408891458716et_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_77_order__refl,axiom,
! [X4: set_set_a] : ( ord_le3724670747650509150_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_78_order__refl,axiom,
! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_79_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_80_order__refl,axiom,
! [X4: a] : ( ord_less_eq_a @ X4 @ X4 ) ).
% order_refl
thf(fact_81_order__refl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_82_interval__def,axiom,
( interval_nat
= ( ^ [I: set_nat] :
! [X3: nat,Y2: nat,Z3: nat] :
( ( member_nat @ X3 @ I )
=> ( ( member_nat @ Z3 @ I )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( member_nat @ Y2 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_83_interval__def,axiom,
( interval_a
= ( ^ [I: set_a] :
! [X3: a,Y2: a,Z3: a] :
( ( member_a @ X3 @ I )
=> ( ( member_a @ Z3 @ I )
=> ( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z3 )
=> ( member_a @ Y2 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_84_closed__atLeast,axiom,
! [A2: nat] : ( topolo517082937297594846ed_nat @ ( set_ord_atLeast_nat @ A2 ) ) ).
% closed_atLeast
thf(fact_85_closed__atLeast,axiom,
! [A2: a] : ( topolo784654279908865136osed_a @ ( set_ord_atLeast_a @ A2 ) ) ).
% closed_atLeast
thf(fact_86_order__le__imp__less__or__eq,axiom,
! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ( ord_less_set_set_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_87_order__le__imp__less__or__eq,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_set_a @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_88_order__le__imp__less__or__eq,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ord_less_set_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_89_order__le__imp__less__or__eq,axiom,
! [X4: a,Y3: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
=> ( ( ord_less_a @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_90_order__le__imp__less__or__eq,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_91_order__le__imp__less__or__eq,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_a @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_92_psubsetD,axiom,
! [A: set_set_set_a,B: set_set_set_a,C: set_set_a] :
( ( ord_le8783759336481291722_set_a @ A @ B )
=> ( ( member_set_set_a @ C @ A )
=> ( member_set_set_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_93_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_94_psubsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_95_psubsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_96_psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_97_psubset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_98_psubset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( ord_less_set_set_a @ B @ C2 )
=> ( ord_less_set_set_a @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_99_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_100_nle__le,axiom,
! [A2: a,B2: a] :
( ( ~ ( ord_less_eq_a @ A2 @ B2 ) )
= ( ( ord_less_eq_a @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_101_le__cases3,axiom,
! [X4: nat,Y3: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_102_le__cases3,axiom,
! [X4: a,Y3: a,Z4: a] :
( ( ( ord_less_eq_a @ X4 @ Y3 )
=> ~ ( ord_less_eq_a @ Y3 @ Z4 ) )
=> ( ( ( ord_less_eq_a @ Y3 @ X4 )
=> ~ ( ord_less_eq_a @ X4 @ Z4 ) )
=> ( ( ( ord_less_eq_a @ X4 @ Z4 )
=> ~ ( ord_less_eq_a @ Z4 @ Y3 ) )
=> ( ( ( ord_less_eq_a @ Z4 @ Y3 )
=> ~ ( ord_less_eq_a @ Y3 @ X4 ) )
=> ( ( ( ord_less_eq_a @ Y3 @ Z4 )
=> ~ ( ord_less_eq_a @ Z4 @ X4 ) )
=> ~ ( ( ord_less_eq_a @ Z4 @ X4 )
=> ~ ( ord_less_eq_a @ X4 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_103_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z5: set_set_nat] : ( Y4 = Z5 ) )
= ( ^ [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
& ( ord_le6893508408891458716et_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_104_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z5: set_set_a] : ( Y4 = Z5 ) )
= ( ^ [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
& ( ord_le3724670747650509150_set_a @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_105_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z5: set_nat] : ( Y4 = Z5 ) )
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
& ( ord_less_eq_set_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_106_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z5: nat] : ( Y4 = Z5 ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_107_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z5: a] : ( Y4 = Z5 ) )
= ( ^ [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
& ( ord_less_eq_a @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_108_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z5: set_a] : ( Y4 = Z5 ) )
= ( ^ [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_109_ord__eq__le__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( A2 = B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_110_ord__eq__le__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A2 = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_111_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_112_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_113_ord__eq__le__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_114_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_115_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_116_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_117_mem__Collect__eq,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( member_set_set_a @ A2 @ ( collect_set_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_118_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_119_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_120_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_121_Collect__mem__eq,axiom,
! [A: set_set_set_a] :
( ( collect_set_set_a
@ ^ [X3: set_set_a] : ( member_set_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_122_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_123_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_124_ord__le__eq__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_125_ord__le__eq__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_126_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_127_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_128_ord__le__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_129_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_130_order__antisym,axiom,
! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ( ord_le6893508408891458716et_nat @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_131_order__antisym,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_132_order__antisym,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_133_order__antisym,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_134_order__antisym,axiom,
! [X4: a,Y3: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_135_order__antisym,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_136_order_Otrans,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 ) ) ) ).
% order.trans
thf(fact_137_order_Otrans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_138_order_Otrans,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 ) ) ) ).
% order.trans
thf(fact_139_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_140_order_Otrans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_141_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_142_order__trans,axiom,
! [X4: set_set_nat,Y3: set_set_nat,Z4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ( ord_le6893508408891458716et_nat @ Y3 @ Z4 )
=> ( ord_le6893508408891458716et_nat @ X4 @ Z4 ) ) ) ).
% order_trans
thf(fact_143_order__trans,axiom,
! [X4: set_set_a,Y3: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ Z4 )
=> ( ord_le3724670747650509150_set_a @ X4 @ Z4 ) ) ) ).
% order_trans
thf(fact_144_order__trans,axiom,
! [X4: set_nat,Y3: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z4 )
=> ( ord_less_eq_set_nat @ X4 @ Z4 ) ) ) ).
% order_trans
thf(fact_145_order__trans,axiom,
! [X4: nat,Y3: nat,Z4: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z4 )
=> ( ord_less_eq_nat @ X4 @ Z4 ) ) ) ).
% order_trans
thf(fact_146_order__trans,axiom,
! [X4: a,Y3: a,Z4: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ Z4 )
=> ( ord_less_eq_a @ X4 @ Z4 ) ) ) ).
% order_trans
thf(fact_147_order__trans,axiom,
! [X4: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z4 )
=> ( ord_less_eq_set_a @ X4 @ Z4 ) ) ) ).
% order_trans
thf(fact_148_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_149_linorder__wlog,axiom,
! [P: a > a > $o,A2: a,B2: a] :
( ! [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_150_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z5: set_set_nat] : ( Y4 = Z5 ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
& ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_151_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_a,Z5: set_set_a] : ( Y4 = Z5 ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_152_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z5: set_nat] : ( Y4 = Z5 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_153_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z5: nat] : ( Y4 = Z5 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_154_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: a,Z5: a] : ( Y4 = Z5 ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ( ord_less_eq_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_155_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z5: set_a] : ( Y4 = Z5 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_156_dual__order_Oantisym,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_157_dual__order_Oantisym,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_158_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_159_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_160_dual__order_Oantisym,axiom,
! [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_161_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_162_dual__order_Otrans,axiom,
! [B2: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ B2 )
=> ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_163_dual__order_Otrans,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_164_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_165_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_166_dual__order_Otrans,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ C @ B2 )
=> ( ord_less_eq_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_167_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_168_antisym,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_169_antisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_170_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_171_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_172_antisym,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_173_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_174_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z5: set_set_nat] : ( Y4 = Z5 ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_175_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z5: set_set_a] : ( Y4 = Z5 ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_176_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z5: set_nat] : ( Y4 = Z5 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_177_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z5: nat] : ( Y4 = Z5 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_178_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z5: a] : ( Y4 = Z5 ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ( ord_less_eq_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_179_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z5: set_a] : ( Y4 = Z5 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_180_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_181_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_182_order__subst1,axiom,
! [A2: nat,F: a > nat,B2: a,C: a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_eq_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_183_order__subst1,axiom,
! [A2: a,F: nat > a,B2: nat,C: nat] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_184_order__subst1,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_eq_a @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_185_order__subst1,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_186_order__subst1,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_eq_a @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_187_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_188_order__subst1,axiom,
! [A2: set_nat,F: a > set_nat,B2: a,C: a] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_eq_a @ X2 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_189_order__subst1,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_190_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_191_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_192_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_193_order__subst2,axiom,
! [A2: a,B2: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_eq_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_194_order__subst2,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_eq_a @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_195_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_196_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_197_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_198_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > a,C: a] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_199_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_200_order__eq__refl,axiom,
! [X4: set_set_nat,Y3: set_set_nat] :
( ( X4 = Y3 )
=> ( ord_le6893508408891458716et_nat @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_201_order__eq__refl,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( X4 = Y3 )
=> ( ord_le3724670747650509150_set_a @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_202_order__eq__refl,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( X4 = Y3 )
=> ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_203_order__eq__refl,axiom,
! [X4: nat,Y3: nat] :
( ( X4 = Y3 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_204_order__eq__refl,axiom,
! [X4: a,Y3: a] :
( ( X4 = Y3 )
=> ( ord_less_eq_a @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_205_order__eq__refl,axiom,
! [X4: set_a,Y3: set_a] :
( ( X4 = Y3 )
=> ( ord_less_eq_set_a @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_206_linorder__linear,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linorder_linear
thf(fact_207_linorder__linear,axiom,
! [X4: a,Y3: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
| ( ord_less_eq_a @ Y3 @ X4 ) ) ).
% linorder_linear
thf(fact_208_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_209_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_210_ord__eq__le__subst,axiom,
! [A2: a,F: nat > a,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_211_ord__eq__le__subst,axiom,
! [A2: nat,F: a > nat,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_eq_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_212_ord__eq__le__subst,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_eq_a @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_213_ord__eq__le__subst,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_214_ord__eq__le__subst,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_215_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_216_ord__eq__le__subst,axiom,
! [A2: a,F: set_nat > a,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_217_ord__eq__le__subst,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_218_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_219_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_220_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_221_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_eq_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_222_ord__le__eq__subst,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_eq_a @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_223_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_224_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_225_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_226_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > a,C: a] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_227_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_228_linorder__le__cases,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linorder_le_cases
thf(fact_229_linorder__le__cases,axiom,
! [X4: a,Y3: a] :
( ~ ( ord_less_eq_a @ X4 @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X4 ) ) ).
% linorder_le_cases
thf(fact_230_order__antisym__conv,axiom,
! [Y3: set_set_nat,X4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y3 @ X4 )
=> ( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_231_order__antisym__conv,axiom,
! [Y3: set_set_a,X4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X4 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_232_order__antisym__conv,axiom,
! [Y3: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_233_order__antisym__conv,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_234_order__antisym__conv,axiom,
! [Y3: a,X4: a] :
( ( ord_less_eq_a @ Y3 @ X4 )
=> ( ( ord_less_eq_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_235_order__antisym__conv,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_236_gt__ex,axiom,
! [X4: nat] :
? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% gt_ex
thf(fact_237_less__imp__neq,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% less_imp_neq
thf(fact_238_less__imp__neq,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% less_imp_neq
thf(fact_239_less__imp__neq,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% less_imp_neq
thf(fact_240_less__imp__neq,axiom,
! [X4: a,Y3: a] :
( ( ord_less_a @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% less_imp_neq
thf(fact_241_less__imp__neq,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% less_imp_neq
thf(fact_242_order_Oasym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_243_order_Oasym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ord_less_set_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_244_order_Oasym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_set_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_245_order_Oasym,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ~ ( ord_less_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_246_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_247_ord__eq__less__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_248_ord__eq__less__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_249_ord__eq__less__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A2 = B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_250_ord__eq__less__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_251_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_252_ord__less__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_253_ord__less__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_254_ord__less__eq__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_255_ord__less__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_256_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_257_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y: nat] :
( ( ord_less_nat @ Y @ X2 )
=> ( P @ Y ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_258_antisym__conv3,axiom,
! [Y3: a,X4: a] :
( ~ ( ord_less_a @ Y3 @ X4 )
=> ( ( ~ ( ord_less_a @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_259_antisym__conv3,axiom,
! [Y3: nat,X4: nat] :
( ~ ( ord_less_nat @ Y3 @ X4 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_260_linorder__cases,axiom,
! [X4: a,Y3: a] :
( ~ ( ord_less_a @ X4 @ Y3 )
=> ( ( X4 != Y3 )
=> ( ord_less_a @ Y3 @ X4 ) ) ) ).
% linorder_cases
thf(fact_261_linorder__cases,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ( X4 != Y3 )
=> ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% linorder_cases
thf(fact_262_dual__order_Oasym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_263_dual__order_Oasym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ~ ( ord_less_set_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_264_dual__order_Oasym,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_set_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_265_dual__order_Oasym,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ~ ( ord_less_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_266_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_267_dual__order_Oirrefl,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_268_dual__order_Oirrefl,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_269_dual__order_Oirrefl,axiom,
! [A2: set_set_a] :
~ ( ord_less_set_set_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_270_dual__order_Oirrefl,axiom,
! [A2: a] :
~ ( ord_less_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_271_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_272_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N: nat] :
( ( P4 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_273_linorder__less__wlog,axiom,
! [P: a > a > $o,A2: a,B2: a] :
( ! [A3: a,B3: a] :
( ( ord_less_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a] : ( P @ A3 @ A3 )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_274_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_275_order_Ostrict__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_276_order_Ostrict__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_277_order_Ostrict__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_278_order_Ostrict__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_279_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_280_not__less__iff__gr__or__eq,axiom,
! [X4: a,Y3: a] :
( ( ~ ( ord_less_a @ X4 @ Y3 ) )
= ( ( ord_less_a @ Y3 @ X4 )
| ( X4 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_281_not__less__iff__gr__or__eq,axiom,
! [X4: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X4 )
| ( X4 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_282_dual__order_Ostrict__trans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_283_dual__order_Ostrict__trans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ( ord_less_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_284_dual__order_Ostrict__trans,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ( ord_less_set_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_285_dual__order_Ostrict__trans,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ( ord_less_a @ C @ B2 )
=> ( ord_less_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_286_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_287_order_Ostrict__implies__not__eq,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_288_order_Ostrict__implies__not__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_289_order_Ostrict__implies__not__eq,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_290_order_Ostrict__implies__not__eq,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_291_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_292_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_293_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_294_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_295_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_296_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_297_linorder__neqE,axiom,
! [X4: a,Y3: a] :
( ( X4 != Y3 )
=> ( ~ ( ord_less_a @ X4 @ Y3 )
=> ( ord_less_a @ Y3 @ X4 ) ) ) ).
% linorder_neqE
thf(fact_298_linorder__neqE,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
=> ( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% linorder_neqE
thf(fact_299_order__less__asym,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X4 ) ) ).
% order_less_asym
thf(fact_300_order__less__asym,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ~ ( ord_less_set_nat @ Y3 @ X4 ) ) ).
% order_less_asym
thf(fact_301_order__less__asym,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y3 )
=> ~ ( ord_less_set_set_a @ Y3 @ X4 ) ) ).
% order_less_asym
thf(fact_302_order__less__asym,axiom,
! [X4: a,Y3: a] :
( ( ord_less_a @ X4 @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X4 ) ) ).
% order_less_asym
thf(fact_303_order__less__asym,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% order_less_asym
thf(fact_304_linorder__neq__iff,axiom,
! [X4: a,Y3: a] :
( ( X4 != Y3 )
= ( ( ord_less_a @ X4 @ Y3 )
| ( ord_less_a @ Y3 @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_305_linorder__neq__iff,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
= ( ( ord_less_nat @ X4 @ Y3 )
| ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_306_order__less__asym_H,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_307_order__less__asym_H,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ord_less_set_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_308_order__less__asym_H,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_set_a @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_309_order__less__asym_H,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ~ ( ord_less_a @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_310_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_311_order__less__trans,axiom,
! [X4: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z4 )
=> ( ord_less_set_a @ X4 @ Z4 ) ) ) ).
% order_less_trans
thf(fact_312_order__less__trans,axiom,
! [X4: set_nat,Y3: set_nat,Z4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ( ord_less_set_nat @ Y3 @ Z4 )
=> ( ord_less_set_nat @ X4 @ Z4 ) ) ) ).
% order_less_trans
thf(fact_313_order__less__trans,axiom,
! [X4: set_set_a,Y3: set_set_a,Z4: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_set_a @ Y3 @ Z4 )
=> ( ord_less_set_set_a @ X4 @ Z4 ) ) ) ).
% order_less_trans
thf(fact_314_order__less__trans,axiom,
! [X4: a,Y3: a,Z4: a] :
( ( ord_less_a @ X4 @ Y3 )
=> ( ( ord_less_a @ Y3 @ Z4 )
=> ( ord_less_a @ X4 @ Z4 ) ) ) ).
% order_less_trans
thf(fact_315_order__less__trans,axiom,
! [X4: nat,Y3: nat,Z4: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z4 )
=> ( ord_less_nat @ X4 @ Z4 ) ) ) ).
% order_less_trans
thf(fact_316_ord__eq__less__subst,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_317_ord__eq__less__subst,axiom,
! [A2: nat,F: a > nat,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_318_ord__eq__less__subst,axiom,
! [A2: a,F: nat > a,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_319_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_320_ord__eq__less__subst,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_321_ord__eq__less__subst,axiom,
! [A2: set_nat,F: a > set_nat,B2: a,C: a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_322_ord__eq__less__subst,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_323_ord__eq__less__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_324_ord__eq__less__subst,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_set_a @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_325_ord__eq__less__subst,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_set_a @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_326_ord__less__eq__subst,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_327_ord__less__eq__subst,axiom,
! [A2: a,B2: a,F: a > nat,C: nat] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_328_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_329_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_330_ord__less__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_331_ord__less__eq__subst,axiom,
! [A2: a,B2: a,F: a > set_nat,C: set_nat] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_332_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_333_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_334_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_set_a @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_335_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_set_a @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_336_order__less__irrefl,axiom,
! [X4: set_a] :
~ ( ord_less_set_a @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_337_order__less__irrefl,axiom,
! [X4: set_nat] :
~ ( ord_less_set_nat @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_338_order__less__irrefl,axiom,
! [X4: set_set_a] :
~ ( ord_less_set_set_a @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_339_order__less__irrefl,axiom,
! [X4: a] :
~ ( ord_less_a @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_340_order__less__irrefl,axiom,
! [X4: nat] :
~ ( ord_less_nat @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_341_order__less__subst1,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_342_order__less__subst1,axiom,
! [A2: a,F: nat > a,B2: nat,C: nat] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_343_order__less__subst1,axiom,
! [A2: nat,F: a > nat,B2: a,C: a] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_344_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_345_order__less__subst1,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_set_a @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_346_order__less__subst1,axiom,
! [A2: a,F: set_nat > a,B2: set_nat,C: set_nat] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ! [X2: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_347_order__less__subst1,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_set_a @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_348_order__less__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ! [X2: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_349_order__less__subst1,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_350_order__less__subst1,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_351_order__less__subst2,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_352_order__less__subst2,axiom,
! [A2: a,B2: a,F: a > nat,C: nat] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_353_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_354_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_355_order__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_356_order__less__subst2,axiom,
! [A2: a,B2: a,F: a > set_nat,C: set_nat] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_357_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_358_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_359_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_set_a @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_360_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_set_a @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_361_order__less__not__sym,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X4 ) ) ).
% order_less_not_sym
thf(fact_362_order__less__not__sym,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ~ ( ord_less_set_nat @ Y3 @ X4 ) ) ).
% order_less_not_sym
thf(fact_363_order__less__not__sym,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y3 )
=> ~ ( ord_less_set_set_a @ Y3 @ X4 ) ) ).
% order_less_not_sym
thf(fact_364_order__less__not__sym,axiom,
! [X4: a,Y3: a] :
( ( ord_less_a @ X4 @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X4 ) ) ).
% order_less_not_sym
thf(fact_365_order__less__not__sym,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% order_less_not_sym
thf(fact_366_order__less__imp__triv,axiom,
! [X4: set_a,Y3: set_a,P: $o] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_367_order__less__imp__triv,axiom,
! [X4: set_nat,Y3: set_nat,P: $o] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ( ord_less_set_nat @ Y3 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_368_order__less__imp__triv,axiom,
! [X4: set_set_a,Y3: set_set_a,P: $o] :
( ( ord_less_set_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_set_a @ Y3 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_369_order__less__imp__triv,axiom,
! [X4: a,Y3: a,P: $o] :
( ( ord_less_a @ X4 @ Y3 )
=> ( ( ord_less_a @ Y3 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_370_order__less__imp__triv,axiom,
! [X4: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_371_linorder__less__linear,axiom,
! [X4: a,Y3: a] :
( ( ord_less_a @ X4 @ Y3 )
| ( X4 = Y3 )
| ( ord_less_a @ Y3 @ X4 ) ) ).
% linorder_less_linear
thf(fact_372_linorder__less__linear,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 )
| ( ord_less_nat @ Y3 @ X4 ) ) ).
% linorder_less_linear
thf(fact_373_order__less__imp__not__eq,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_374_order__less__imp__not__eq,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_375_order__less__imp__not__eq,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_376_order__less__imp__not__eq,axiom,
! [X4: a,Y3: a] :
( ( ord_less_a @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_377_order__less__imp__not__eq,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_378_order__less__imp__not__eq2,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( Y3 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_379_order__less__imp__not__eq2,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( Y3 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_380_order__less__imp__not__eq2,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y3 )
=> ( Y3 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_381_order__less__imp__not__eq2,axiom,
! [X4: a,Y3: a] :
( ( ord_less_a @ X4 @ Y3 )
=> ( Y3 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_382_order__less__imp__not__eq2,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( Y3 != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_383_order__less__imp__not__less,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_384_order__less__imp__not__less,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ~ ( ord_less_set_nat @ Y3 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_385_order__less__imp__not__less,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y3 )
=> ~ ( ord_less_set_set_a @ Y3 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_386_order__less__imp__not__less,axiom,
! [X4: a,Y3: a] :
( ( ord_less_a @ X4 @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_387_order__less__imp__not__less,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_388_in__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a,X4: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( member_set_set_a @ X4 @ A )
=> ( member_set_set_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_389_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X4: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_390_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X4: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X4 @ A )
=> ( member_set_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_391_in__mono,axiom,
! [A: set_nat,B: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_392_in__mono,axiom,
! [A: set_a,B: set_a,X4: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_393_subsetD,axiom,
! [A: set_set_set_a,B: set_set_set_a,C: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( member_set_set_a @ C @ A )
=> ( member_set_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_394_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_395_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_396_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_397_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_398_equalityE,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ~ ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_399_equalityE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_400_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_401_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_402_subset__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A5: set_set_set_a,B5: set_set_set_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A5 )
=> ( member_set_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_403_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A5 )
=> ( member_set_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_404_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( member_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_405_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_406_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_407_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_408_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_409_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_410_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_411_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_412_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_413_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_414_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_415_subset__iff,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A5: set_set_set_a,B5: set_set_set_a] :
! [T2: set_set_a] :
( ( member_set_set_a @ T2 @ A5 )
=> ( member_set_set_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_416_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [T2: set_nat] :
( ( member_set_nat @ T2 @ A5 )
=> ( member_set_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_417_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A5 )
=> ( member_set_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_418_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_419_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A5 )
=> ( member_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_420_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_421_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_422_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_423_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_424_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X2: set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_425_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_426_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_427_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_428_subset__trans,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 ) ) ) ).
% subset_trans
thf(fact_429_subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_430_subset__trans,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 ) ) ) ).
% subset_trans
thf(fact_431_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_432_set__eq__subset,axiom,
( ( ^ [Y4: set_set_nat,Z5: set_set_nat] : ( Y4 = Z5 ) )
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B5 )
& ( ord_le6893508408891458716et_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_433_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z5: set_set_a] : ( Y4 = Z5 ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_434_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z5: set_nat] : ( Y4 = Z5 ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_435_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z5: set_a] : ( Y4 = Z5 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_436_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_437_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_438_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_439_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_440_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_less_set_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_441_subset__iff__psubset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_less_set_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_442_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_443_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_444_subset__psubset__trans,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 ) ) ) ).
% subset_psubset_trans
thf(fact_445_subset__psubset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_set_set_a @ B @ C2 )
=> ( ord_less_set_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_446_subset__psubset__trans,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 ) ) ) ).
% subset_psubset_trans
thf(fact_447_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_448_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B5 )
& ~ ( ord_le6893508408891458716et_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_449_subset__not__subset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ~ ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_450_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_451_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_452_psubset__subset__trans,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 ) ) ) ).
% psubset_subset_trans
thf(fact_453_psubset__subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_less_set_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_454_psubset__subset__trans,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 ) ) ) ).
% psubset_subset_trans
thf(fact_455_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_456_psubset__imp__subset,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_457_psubset__imp__subset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_458_psubset__imp__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_459_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_460_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_461_psubset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_462_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_463_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_464_psubsetE,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_465_psubsetE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_466_psubsetE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_467_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_468_leD,axiom,
! [Y3: set_set_nat,X4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y3 @ X4 )
=> ~ ( ord_less_set_set_nat @ X4 @ Y3 ) ) ).
% leD
thf(fact_469_leD,axiom,
! [Y3: set_set_a,X4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X4 )
=> ~ ( ord_less_set_set_a @ X4 @ Y3 ) ) ).
% leD
thf(fact_470_leD,axiom,
! [Y3: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X4 )
=> ~ ( ord_less_set_nat @ X4 @ Y3 ) ) ).
% leD
thf(fact_471_leD,axiom,
! [Y3: a,X4: a] :
( ( ord_less_eq_a @ Y3 @ X4 )
=> ~ ( ord_less_a @ X4 @ Y3 ) ) ).
% leD
thf(fact_472_leD,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ~ ( ord_less_nat @ X4 @ Y3 ) ) ).
% leD
thf(fact_473_leD,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ~ ( ord_less_set_a @ X4 @ Y3 ) ) ).
% leD
thf(fact_474_leI,axiom,
! [X4: a,Y3: a] :
( ~ ( ord_less_a @ X4 @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X4 ) ) ).
% leI
thf(fact_475_leI,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% leI
thf(fact_476_nless__le,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ~ ( ord_less_set_set_nat @ A2 @ B2 ) )
= ( ~ ( ord_le6893508408891458716et_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_477_nless__le,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ~ ( ord_less_set_set_a @ A2 @ B2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_478_nless__le,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_479_nless__le,axiom,
! [A2: a,B2: a] :
( ( ~ ( ord_less_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_480_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_481_nless__le,axiom,
! [A2: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_482_antisym__conv1,axiom,
! [X4: set_set_nat,Y3: set_set_nat] :
( ~ ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_483_antisym__conv1,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ~ ( ord_less_set_set_a @ X4 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_484_antisym__conv1,axiom,
! [X4: set_nat,Y3: set_nat] :
( ~ ( ord_less_set_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_485_antisym__conv1,axiom,
! [X4: a,Y3: a] :
( ~ ( ord_less_a @ X4 @ Y3 )
=> ( ( ord_less_eq_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_486_antisym__conv1,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_487_antisym__conv1,axiom,
! [X4: set_a,Y3: set_a] :
( ~ ( ord_less_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_488_antisym__conv2,axiom,
! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ( ~ ( ord_less_set_set_nat @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_489_antisym__conv2,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ( ~ ( ord_less_set_set_a @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_490_antisym__conv2,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ~ ( ord_less_set_nat @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_491_antisym__conv2,axiom,
! [X4: a,Y3: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
=> ( ( ~ ( ord_less_a @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_492_antisym__conv2,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_493_antisym__conv2,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ~ ( ord_less_set_a @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_494_less__le__not__le,axiom,
( ord_less_set_set_nat
= ( ^ [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
& ~ ( ord_le6893508408891458716et_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_495_less__le__not__le,axiom,
( ord_less_set_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
& ~ ( ord_le3724670747650509150_set_a @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_496_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
& ~ ( ord_less_eq_set_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_497_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
& ~ ( ord_less_eq_a @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_498_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_499_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
& ~ ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_500_not__le__imp__less,axiom,
! [Y3: a,X4: a] :
( ~ ( ord_less_eq_a @ Y3 @ X4 )
=> ( ord_less_a @ X4 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_501_not__le__imp__less,axiom,
! [Y3: nat,X4: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ord_less_nat @ X4 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_502_order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_less_set_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_503_order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_less_set_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_504_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_505_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_506_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_507_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_508_order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_509_order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_510_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_511_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_512_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_513_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_514_order_Ostrict__trans1,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 ) ) ) ).
% order.strict_trans1
thf(fact_515_order_Ostrict__trans1,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_516_order_Ostrict__trans1,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 ) ) ) ).
% order.strict_trans1
thf(fact_517_order_Ostrict__trans1,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_518_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_519_order_Ostrict__trans1,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_520_order_Ostrict__trans2,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 ) ) ) ).
% order.strict_trans2
thf(fact_521_order_Ostrict__trans2,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_522_order_Ostrict__trans2,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 ) ) ) ).
% order.strict_trans2
thf(fact_523_order_Ostrict__trans2,axiom,
! [A2: a,B2: a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ B2 @ C )
=> ( ord_less_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_524_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_525_order_Ostrict__trans2,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_526_order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ~ ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_527_order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ~ ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_528_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_529_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ~ ( ord_less_eq_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_530_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_531_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_532_dual__order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( ord_less_set_set_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_533_dual__order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_less_set_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_534_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_535_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_536_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_537_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_538_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_539_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_540_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_541_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_542_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_543_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_544_dual__order_Ostrict__trans1,axiom,
! [B2: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( ord_less_set_set_nat @ C @ B2 )
=> ( ord_less_set_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_545_dual__order_Ostrict__trans1,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_less_set_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_546_dual__order_Ostrict__trans1,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_547_dual__order_Ostrict__trans1,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_eq_a @ B2 @ A2 )
=> ( ( ord_less_a @ C @ B2 )
=> ( ord_less_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_548_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_549_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_550_dual__order_Ostrict__trans2,axiom,
! [B2: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ B2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ B2 )
=> ( ord_less_set_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_551_dual__order_Ostrict__trans2,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_552_dual__order_Ostrict__trans2,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_553_dual__order_Ostrict__trans2,axiom,
! [B2: a,A2: a,C: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ( ord_less_eq_a @ C @ B2 )
=> ( ord_less_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_554_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_555_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_556_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
& ~ ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_557_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ~ ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_558_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_559_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B4: a,A4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ~ ( ord_less_eq_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_560_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_561_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_562_order_Ostrict__implies__order,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_563_order_Ostrict__implies__order,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_564_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_565_order_Ostrict__implies__order,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ord_less_eq_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_566_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_567_order_Ostrict__implies__order,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_568_dual__order_Ostrict__implies__order,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ B2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_569_dual__order_Ostrict__implies__order,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_570_dual__order_Ostrict__implies__order,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_571_dual__order_Ostrict__implies__order,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ord_less_eq_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_572_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_573_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_574_order__le__less,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_less_set_set_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_575_order__le__less,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] :
( ( ord_less_set_set_a @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_576_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_577_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X3: a,Y2: a] :
( ( ord_less_a @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_578_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_579_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_580_order__less__le,axiom,
( ord_less_set_set_nat
= ( ^ [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_581_order__less__le,axiom,
( ord_less_set_set_a
= ( ^ [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_582_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_583_order__less__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_584_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_585_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_586_linorder__not__le,axiom,
! [X4: a,Y3: a] :
( ( ~ ( ord_less_eq_a @ X4 @ Y3 ) )
= ( ord_less_a @ Y3 @ X4 ) ) ).
% linorder_not_le
thf(fact_587_linorder__not__le,axiom,
! [X4: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X4 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X4 ) ) ).
% linorder_not_le
thf(fact_588_linorder__not__less,axiom,
! [X4: a,Y3: a] :
( ( ~ ( ord_less_a @ X4 @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X4 ) ) ).
% linorder_not_less
thf(fact_589_linorder__not__less,axiom,
! [X4: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linorder_not_less
thf(fact_590_order__less__imp__le,axiom,
! [X4: set_set_nat,Y3: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ X4 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_591_order__less__imp__le,axiom,
! [X4: set_set_a,Y3: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ X4 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_592_order__less__imp__le,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_593_order__less__imp__le,axiom,
! [X4: a,Y3: a] :
( ( ord_less_a @ X4 @ Y3 )
=> ( ord_less_eq_a @ X4 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_594_order__less__imp__le,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_595_order__less__imp__le,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ X4 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_596_order__le__neq__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_597_order__le__neq__trans,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_598_order__le__neq__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_599_order__le__neq__trans,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_600_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_601_order__le__neq__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_602_order__neq__le__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 != B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_603_order__neq__le__trans,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 != B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_604_order__neq__le__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_605_order__neq__le__trans,axiom,
! [A2: a,B2: a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_a @ A2 @ B2 )
=> ( ord_less_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_606_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_607_order__neq__le__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_608_order__le__less__trans,axiom,
! [X4: set_set_nat,Y3: set_set_nat,Z4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
=> ( ( ord_less_set_set_nat @ Y3 @ Z4 )
=> ( ord_less_set_set_nat @ X4 @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_609_order__le__less__trans,axiom,
! [X4: set_set_a,Y3: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_set_a @ Y3 @ Z4 )
=> ( ord_less_set_set_a @ X4 @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_610_order__le__less__trans,axiom,
! [X4: set_nat,Y3: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ord_less_set_nat @ Y3 @ Z4 )
=> ( ord_less_set_nat @ X4 @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_611_order__le__less__trans,axiom,
! [X4: a,Y3: a,Z4: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
=> ( ( ord_less_a @ Y3 @ Z4 )
=> ( ord_less_a @ X4 @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_612_order__le__less__trans,axiom,
! [X4: nat,Y3: nat,Z4: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z4 )
=> ( ord_less_nat @ X4 @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_613_order__le__less__trans,axiom,
! [X4: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z4 )
=> ( ord_less_set_a @ X4 @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_614_order__less__le__trans,axiom,
! [X4: set_set_nat,Y3: set_set_nat,Z4: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ Y3 )
=> ( ( ord_le6893508408891458716et_nat @ Y3 @ Z4 )
=> ( ord_less_set_set_nat @ X4 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_615_order__less__le__trans,axiom,
! [X4: set_set_a,Y3: set_set_a,Z4: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ Z4 )
=> ( ord_less_set_set_a @ X4 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_616_order__less__le__trans,axiom,
! [X4: set_nat,Y3: set_nat,Z4: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z4 )
=> ( ord_less_set_nat @ X4 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_617_order__less__le__trans,axiom,
! [X4: a,Y3: a,Z4: a] :
( ( ord_less_a @ X4 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ Z4 )
=> ( ord_less_a @ X4 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_618_order__less__le__trans,axiom,
! [X4: nat,Y3: nat,Z4: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z4 )
=> ( ord_less_nat @ X4 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_619_order__less__le__trans,axiom,
! [X4: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z4 )
=> ( ord_less_set_a @ X4 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_620_order__le__less__subst1,axiom,
! [A2: a,F: set_set_a > a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ! [X2: set_set_a,Y5: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_621_order__le__less__subst1,axiom,
! [A2: a,F: a > a,B2: a,C: a] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_622_order__le__less__subst1,axiom,
! [A2: nat,F: a > nat,B2: a,C: a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_623_order__le__less__subst1,axiom,
! [A2: a,F: nat > a,B2: nat,C: nat] :
( ( ord_less_eq_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_624_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_625_order__le__less__subst1,axiom,
! [A2: set_a,F: a > set_a,B2: a,C: a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a @ B2 @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_626_order__le__less__subst1,axiom,
! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_627_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_628_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_629_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_630_order__less__le__subst1,axiom,
! [A2: a,F: set_a > a,B2: set_a,C: set_a] :
( ( ord_less_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_631_order__less__le__subst1,axiom,
! [A2: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_632_order__less__le__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_633_order__less__le__subst2,axiom,
! [A2: a,B2: a,F: a > a,C: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_634_order__less__le__subst2,axiom,
! [A2: a,B2: a,F: a > nat,C: nat] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_635_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_a @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_636_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_637_order__less__le__subst2,axiom,
! [A2: a,B2: a,F: a > set_a,C: set_a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a,Y5: a] :
( ( ord_less_a @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_638_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_639_linorder__le__less__linear,axiom,
! [X4: a,Y3: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
| ( ord_less_a @ Y3 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_640_linorder__le__less__linear,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
| ( ord_less_nat @ Y3 @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_641_verit__comp__simplify1_I3_J,axiom,
! [B6: a,A6: a] :
( ( ~ ( ord_less_eq_a @ B6 @ A6 ) )
= ( ord_less_a @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_642_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_643_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B2 )
& ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ C3 ) )
=> ( P @ X ) )
& ! [D: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A2 @ X2 )
& ( ord_less_nat @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_644_GreatestI2__order,axiom,
! [P: set_a > $o,X4: set_a,Q: set_a > $o] :
( ( P @ X4 )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X4 ) )
=> ( ! [X2: set_a] :
( ( P @ X2 )
=> ( ! [Y: set_a] :
( ( P @ Y )
=> ( ord_less_eq_set_a @ Y @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_645_Greatest__equality,axiom,
! [P: set_a > $o,X4: set_a] :
( ( P @ X4 )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X4 ) )
=> ( ( order_Greatest_set_a @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_646_Icc__subset__Ici__iff,axiom,
! [L: set_a,H: set_a,L2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ L @ H ) @ ( set_or8362275514725411625_set_a @ L2 ) )
= ( ~ ( ord_less_eq_set_a @ L @ H )
| ( ord_less_eq_set_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_647_Icc__subset__Ici__iff,axiom,
! [L: a,H: a,L2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ L @ H ) @ ( set_ord_atLeast_a @ L2 ) )
= ( ~ ( ord_less_eq_a @ L @ H )
| ( ord_less_eq_a @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_648_Icc__subset__Ici__iff,axiom,
! [L: nat,H: nat,L2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atLeast_nat @ L2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_649_verit__comp__simplify1_I1_J,axiom,
! [A2: a] :
~ ( ord_less_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_650_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_651_atLeastAtMost__iff,axiom,
! [I2: a,L: a,U: a] :
( ( member_a @ I2 @ ( set_or672772299803893939Most_a @ L @ U ) )
= ( ( ord_less_eq_a @ L @ I2 )
& ( ord_less_eq_a @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_652_atLeastAtMost__iff,axiom,
! [I2: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I2 @ ( set_or6288561110385358355_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_653_atLeastAtMost__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_654_Icc__eq__Icc,axiom,
! [L: set_a,H: set_a,L2: set_a,H2: set_a] :
( ( ( set_or6288561110385358355_set_a @ L @ H )
= ( set_or6288561110385358355_set_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_a @ L @ H )
& ~ ( ord_less_eq_set_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_655_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_656_atLeastatMost__subset__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or6288561110385358355_set_a @ C @ D2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_657_atLeastatMost__subset__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or672772299803893939Most_a @ C @ D2 ) )
= ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_658_atLeastatMost__subset__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_659_closed__atLeastAtMost,axiom,
! [A2: nat,B2: nat] : ( topolo517082937297594846ed_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) ) ).
% closed_atLeastAtMost
thf(fact_660_closed__atLeastAtMost,axiom,
! [A2: a,B2: a] : ( topolo784654279908865136osed_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) ) ).
% closed_atLeastAtMost
thf(fact_661_not__Ici__eq__Icc,axiom,
! [L2: nat,L: nat,H: nat] :
( ( set_ord_atLeast_nat @ L2 )
!= ( set_or1269000886237332187st_nat @ L @ H ) ) ).
% not_Ici_eq_Icc
thf(fact_662_not__Ici__le__Icc,axiom,
! [L: nat,L2: nat,H2: nat] :
~ ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ L ) @ ( set_or1269000886237332187st_nat @ L2 @ H2 ) ) ).
% not_Ici_le_Icc
thf(fact_663_atLeastatMost__psubset__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ord_less_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or672772299803893939Most_a @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_a @ A2 @ B2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D2 )
& ( ( ord_less_a @ C @ A2 )
| ( ord_less_a @ B2 @ D2 ) ) ) )
& ( ord_less_eq_a @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_664_atLeastatMost__psubset__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D2: set_a] :
( ( ord_less_set_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or6288561110385358355_set_a @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ B2 @ D2 )
& ( ( ord_less_set_a @ C @ A2 )
| ( ord_less_set_a @ B2 @ D2 ) ) ) )
& ( ord_less_eq_set_a @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_665_atLeastatMost__psubset__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B2 @ D2 )
& ( ( ord_less_nat @ C @ A2 )
| ( ord_less_nat @ B2 @ D2 ) ) ) )
& ( ord_less_eq_nat @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_666_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_667_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A2 @ B2 ) @ ( set_or2348907005316661231_set_a @ C @ D2 ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_set_a @ B2 @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_668_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( set_or5139330845457685135Than_a @ C @ D2 ) )
= ( ( ord_less_eq_a @ A2 @ B2 )
=> ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_a @ B2 @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_669_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_nat @ B2 @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_670_Icc__subset__Iic__iff,axiom,
! [L: set_a,H: set_a,H2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ L @ H ) @ ( set_ord_atMost_set_a @ H2 ) )
= ( ~ ( ord_less_eq_set_a @ L @ H )
| ( ord_less_eq_set_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_671_Icc__subset__Iic__iff,axiom,
! [L: a,H: a,H2: a] :
( ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ L @ H ) @ ( set_ord_atMost_a @ H2 ) )
= ( ~ ( ord_less_eq_a @ L @ H )
| ( ord_less_eq_a @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_672_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_673_chain__subset__def,axiom,
( chain_subset_a
= ( ^ [C4: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ C4 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ C4 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
| ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_674_atMost__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_atMost_a @ K ) )
= ( ord_less_eq_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_675_atMost__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_ord_atMost_set_a @ K ) )
= ( ord_less_eq_set_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_676_greaterThanLessThan__iff,axiom,
! [I2: a,L: a,U: a] :
( ( member_a @ I2 @ ( set_or5939364468397584554Than_a @ L @ U ) )
= ( ( ord_less_a @ L @ I2 )
& ( ord_less_a @ I2 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_677_greaterThanLessThan__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I2 )
& ( ord_less_nat @ I2 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_678_atLeastLessThan__iff,axiom,
! [I2: a,L: a,U: a] :
( ( member_a @ I2 @ ( set_or5139330845457685135Than_a @ L @ U ) )
= ( ( ord_less_eq_a @ L @ I2 )
& ( ord_less_a @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_679_atLeastLessThan__iff,axiom,
! [I2: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I2 @ ( set_or2348907005316661231_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_set_a @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_680_atLeastLessThan__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_nat @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_681_ivl__subset,axiom,
! [I2: a,J: a,M2: a,N2: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ I2 @ J ) @ ( set_or5139330845457685135Than_a @ M2 @ N2 ) )
= ( ( ord_less_eq_a @ J @ I2 )
| ( ( ord_less_eq_a @ M2 @ I2 )
& ( ord_less_eq_a @ J @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_682_ivl__subset,axiom,
! [I2: nat,J: nat,M2: nat,N2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ J @ I2 )
| ( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_eq_nat @ J @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_683_atMost__subset__iff,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X4 ) @ ( set_ord_atMost_set_a @ Y3 ) )
= ( ord_less_eq_set_a @ X4 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_684_atMost__subset__iff,axiom,
! [X4: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ X4 ) @ ( set_ord_atMost_a @ Y3 ) )
= ( ord_less_eq_a @ X4 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_685_greaterThanAtMost__iff,axiom,
! [I2: a,L: a,U: a] :
( ( member_a @ I2 @ ( set_or4472690218693186638Most_a @ L @ U ) )
= ( ( ord_less_a @ L @ I2 )
& ( ord_less_eq_a @ I2 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_686_greaterThanAtMost__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I2 )
& ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_687_greaterThanAtMost__iff,axiom,
! [I2: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I2 @ ( set_or2503527069484367278_set_a @ L @ U ) )
= ( ( ord_less_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_688_atLeastLessThan__inj_I2_J,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D2 ) )
=> ( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ C @ D2 )
=> ( B2 = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_689_atLeastLessThan__inj_I2_J,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( B2 = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_690_atLeastLessThan__inj_I1_J,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D2 ) )
=> ( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ C @ D2 )
=> ( A2 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_691_atLeastLessThan__inj_I1_J,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( A2 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_692_Ico__eq__Ico,axiom,
! [L: a,H: a,L2: a,H2: a] :
( ( ( set_or5139330845457685135Than_a @ L @ H )
= ( set_or5139330845457685135Than_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_a @ L @ H )
& ~ ( ord_less_a @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_693_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_694_atLeastLessThan__eq__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( ord_less_a @ C @ D2 )
=> ( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= ( set_or5139330845457685135Than_a @ C @ D2 ) )
= ( ( A2 = C )
& ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_695_atLeastLessThan__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( A2 = C )
& ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_696_closed__atMost,axiom,
! [A2: a] : ( topolo784654279908865136osed_a @ ( set_ord_atMost_a @ A2 ) ) ).
% closed_atMost
thf(fact_697_atLeastLessThan__subset__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ A2 @ B2 ) @ ( set_or5139330845457685135Than_a @ C @ D2 ) )
=> ( ( ord_less_eq_a @ B2 @ A2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_698_atLeastLessThan__subset__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B2 @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_699_Ioc__subset__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ A2 @ B2 ) @ ( set_or4472690218693186638Most_a @ C @ D2 ) )
= ( ( ord_less_eq_a @ B2 @ A2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_700_ivl__disj__un__two_I5_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_701_ivl__disj__un__two_I5_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_702_ivl__disj__un__two_I4_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or5939364468397584554Than_a @ M2 @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_703_ivl__disj__un__two_I4_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or5834768355832116004an_nat @ M2 @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_704_ivl__disj__un__two__touch_I1_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_705_ivl__disj__un__two__touch_I1_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_nat @ L @ M2 )
=> ( ( ord_less_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_706_ivl__disj__un__two_I2_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or5939364468397584554Than_a @ M2 @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_707_ivl__disj__un__two_I2_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M2 ) @ ( set_or5834768355832116004an_nat @ M2 @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_708_ivl__disj__un__two_I1_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= ( set_or5939364468397584554Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_709_ivl__disj__un__two_I1_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_710_ivl__disj__un__two__touch_I3_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_a @ L @ M2 )
=> ( ( ord_less_eq_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or4472690218693186638Most_a @ L @ M2 ) @ ( set_or672772299803893939Most_a @ M2 @ U ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_711_ivl__disj__un__two__touch_I3_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_712_ivl__disj__un__two__touch_I2_J,axiom,
! [L: a,M2: a,U: a] :
( ( ord_less_eq_a @ L @ M2 )
=> ( ( ord_less_a @ M2 @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ M2 ) @ ( set_or5139330845457685135Than_a @ M2 @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_713_ivl__disj__un__two__touch_I2_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_714_atLeastAtMost__def,axiom,
( set_or672772299803893939Most_a
= ( ^ [L3: a,U2: a] : ( inf_inf_set_a @ ( set_ord_atLeast_a @ L3 ) @ ( set_ord_atMost_a @ U2 ) ) ) ) ).
% atLeastAtMost_def
thf(fact_715_atLeastAtMost__def,axiom,
( set_or1269000886237332187st_nat
= ( ^ [L3: nat,U2: nat] : ( inf_inf_set_nat @ ( set_ord_atLeast_nat @ L3 ) @ ( set_ord_atMost_nat @ U2 ) ) ) ) ).
% atLeastAtMost_def
thf(fact_716_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_717_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_718_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_719_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_720_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_721_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_722_closed__Int,axiom,
! [S: set_a,T3: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( topolo784654279908865136osed_a @ ( inf_inf_set_a @ S @ T3 ) ) ) ) ).
% closed_Int
thf(fact_723_closed__Un,axiom,
! [S: set_a,T3: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( topolo784654279908865136osed_a @ ( sup_sup_set_a @ S @ T3 ) ) ) ) ).
% closed_Un
thf(fact_724_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_725_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_726_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_727_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_728_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_729_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_730_Un__Int__assoc__eq,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_731_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_732_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B )
=> ( C2
!= ( sup_sup_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_733_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_734_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_735_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_736_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_737_Un__least,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_738_Un__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D3: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_739_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_740_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_741_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_742_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_743_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_744_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_745_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D3: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D3 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_746_ivl__disj__un__two__touch_I4_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_747_ivl__disj__un__two_I3_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_748_ivl__disj__un__two_I7_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_749_ivl__disj__un__two_I8_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or6659071591806873216st_nat @ M2 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_750_ivl__disj__un__one_I8_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or5139330845457685135Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= ( set_ord_atLeast_a @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_751_ivl__disj__un__one_I8_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U ) @ ( set_ord_atLeast_nat @ U ) )
= ( set_ord_atLeast_nat @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_752_le__sup__iff,axiom,
! [X4: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X4 @ Y3 ) @ Z4 )
= ( ( ord_less_eq_set_a @ X4 @ Z4 )
& ( ord_less_eq_set_a @ Y3 @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_753_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_754_le__inf__iff,axiom,
! [X4: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z4 ) )
= ( ( ord_less_eq_set_a @ X4 @ Y3 )
& ( ord_less_eq_set_a @ X4 @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_755_inf_Obounded__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_756_distrib__inf__le,axiom,
! [X4: set_a,Y3: set_a,Z4: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ ( inf_inf_set_a @ X4 @ Z4 ) ) @ ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y3 @ Z4 ) ) ) ).
% distrib_inf_le
thf(fact_757_distrib__sup__le,axiom,
! [X4: set_a,Y3: set_a,Z4: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z4 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y3 ) @ ( sup_sup_set_a @ X4 @ Z4 ) ) ) ).
% distrib_sup_le
thf(fact_758_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_759_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_760_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_761_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_762_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_763_inf_Ocobounded2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_764_inf_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_765_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_766_inf__greatest,axiom,
! [X4: set_a,Y3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ X4 @ Z4 )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z4 ) ) ) ) ).
% inf_greatest
thf(fact_767_inf_OboundedI,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_768_inf_OboundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_769_inf__absorb2,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_770_inf__absorb1,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= X4 ) ) ).
% inf_absorb1
thf(fact_771_inf_Oabsorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_772_inf_Oabsorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_773_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_774_inf__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y3: set_a] :
( ! [X2: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y5 ) @ X2 )
=> ( ! [X2: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y5 ) @ Y5 )
=> ( ! [X2: set_a,Y5: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ( ord_less_eq_set_a @ X2 @ Z )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y5 @ Z ) ) ) )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_775_inf_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_776_inf_OorderE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_777_le__infI2,axiom,
! [B2: set_a,X4: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X4 ) ) ).
% le_infI2
thf(fact_778_le__infI1,axiom,
! [A2: set_a,X4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X4 ) ) ).
% le_infI1
thf(fact_779_inf__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_780_le__infI,axiom,
! [X4: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X4 @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ B2 )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_781_le__infE,axiom,
! [X4: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X4 @ A2 )
=> ~ ( ord_less_eq_set_a @ X4 @ B2 ) ) ) ).
% le_infE
thf(fact_782_inf__le2,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_783_inf__le1,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ X4 ) ).
% inf_le1
thf(fact_784_inf__sup__ord_I1_J,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_785_inf__sup__ord_I2_J,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_786_less__infI1,axiom,
! [A2: nat,X4: nat,B2: nat] :
( ( ord_less_nat @ A2 @ X4 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X4 ) ) ).
% less_infI1
thf(fact_787_less__infI2,axiom,
! [B2: nat,X4: nat,A2: nat] :
( ( ord_less_nat @ B2 @ X4 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X4 ) ) ).
% less_infI2
thf(fact_788_inf_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_789_inf_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_790_inf_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_791_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_792_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ A2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_793_inf_Ostrict__coboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_794_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_795_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_796_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_797_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_798_sup_Ocobounded2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_799_sup_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_800_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_801_sup_OboundedI,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_802_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_803_sup__absorb2,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( sup_sup_set_a @ X4 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_804_sup__absorb1,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( sup_sup_set_a @ X4 @ Y3 )
= X4 ) ) ).
% sup_absorb1
thf(fact_805_sup_Oabsorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_806_sup_Oabsorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_807_sup__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y3: set_a] :
( ! [X2: set_a,Y5: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y5 ) )
=> ( ! [X2: set_a,Y5: set_a] : ( ord_less_eq_set_a @ Y5 @ ( F @ X2 @ Y5 ) )
=> ( ! [X2: set_a,Y5: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y5 @ X2 )
=> ( ( ord_less_eq_set_a @ Z @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y5 @ Z ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_808_sup_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_809_sup_OorderE,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_810_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_811_sup__least,axiom,
! [Y3: set_a,X4: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( ord_less_eq_set_a @ Z4 @ X4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y3 @ Z4 ) @ X4 ) ) ) ).
% sup_least
thf(fact_812_sup__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_813_sup_Omono,axiom,
! [C: set_a,A2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_814_le__supI2,axiom,
! [X4: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X4 @ B2 )
=> ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_815_le__supI1,axiom,
! [X4: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_816_sup__ge2,axiom,
! [Y3: set_a,X4: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X4 @ Y3 ) ) ).
% sup_ge2
thf(fact_817_sup__ge1,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y3 ) ) ).
% sup_ge1
thf(fact_818_le__supI,axiom,
! [A2: set_a,X4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X4 )
=> ( ( ord_less_eq_set_a @ B2 @ X4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_819_le__supE,axiom,
! [A2: set_a,B2: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X4 )
=> ~ ( ord_less_eq_set_a @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_820_inf__sup__ord_I3_J,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_821_inf__sup__ord_I4_J,axiom,
! [Y3: set_a,X4: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X4 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_822_less__supI1,axiom,
! [X4: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ X4 @ A2 )
=> ( ord_less_nat @ X4 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_823_less__supI2,axiom,
! [X4: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ X4 @ B2 )
=> ( ord_less_nat @ X4 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_824_sup_Oabsorb3,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_825_sup_Oabsorb4,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_826_sup_Ostrict__boundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_827_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_828_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_829_ivl__disj__un__one_I6_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ U ) @ ( set_ord_atLeast_nat @ U ) )
= ( set_or1210151606488870762an_nat @ L ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_830_ivl__disj__un__one_I6_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= ( set_or8632414552788122084Than_a @ L ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_831_ivl__disj__un__one_I1_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_ord_atMost_a @ L ) @ ( set_or5939364468397584554Than_a @ L @ U ) )
= ( set_ord_lessThan_a @ U ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_832_ivl__disj__un__one_I1_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( set_ord_lessThan_nat @ U ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_833_ivl__disj__un__one_I7_J,axiom,
! [L: a,U: a] :
( ( ord_less_eq_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or672772299803893939Most_a @ L @ U ) @ ( set_or8632414552788122084Than_a @ U ) )
= ( set_ord_atLeast_a @ L ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_834_ivl__disj__un__one_I7_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ U ) @ ( set_or1210151606488870762an_nat @ U ) )
= ( set_ord_atLeast_nat @ L ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_835_ivl__disj__un__one_I4_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( set_ord_atMost_nat @ U ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_836_ivl__disj__int__one_I6_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_or5939364468397584554Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(6)
thf(fact_837_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_838_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_839_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_840_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_841_lessThan__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_lessThan_a @ K ) )
= ( ord_less_a @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_842_lessThan__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_843_closed__empty,axiom,
topolo784654279908865136osed_a @ bot_bot_set_a ).
% closed_empty
thf(fact_844_greaterThan__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_or8632414552788122084Than_a @ K ) )
= ( ord_less_a @ K @ I2 ) ) ).
% greaterThan_iff
thf(fact_845_greaterThan__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_or1210151606488870762an_nat @ K ) )
= ( ord_less_nat @ K @ I2 ) ) ).
% greaterThan_iff
thf(fact_846_atLeastatMost__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( set_or6288561110385358355_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_847_atLeastatMost__empty__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_848_atLeastatMost__empty__iff2,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_set_a
= ( set_or6288561110385358355_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_849_atLeastatMost__empty__iff2,axiom,
! [A2: nat,B2: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_850_atLeastatMost__empty_H,axiom,
! [A2: set_a,B2: set_a] :
( ~ ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( set_or6288561110385358355_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% atLeastatMost_empty'
thf(fact_851_atLeastatMost__empty_H,axiom,
! [A2: nat,B2: nat] :
( ~ ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty'
thf(fact_852_atLeastatMost__empty,axiom,
! [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
=> ( ( set_or672772299803893939Most_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% atLeastatMost_empty
thf(fact_853_atLeastatMost__empty,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_854_lessThan__subset__iff,axiom,
! [X4: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_ord_lessThan_a @ X4 ) @ ( set_ord_lessThan_a @ Y3 ) )
= ( ord_less_eq_a @ X4 @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_855_atLeastLessThan__empty,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( set_or2348907005316661231_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% atLeastLessThan_empty
thf(fact_856_atLeastLessThan__empty,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_857_atLeastLessThan__empty__iff,axiom,
! [A2: a,B2: a] :
( ( ( set_or5139330845457685135Than_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ~ ( ord_less_a @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_858_atLeastLessThan__empty__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_859_atLeastLessThan__empty__iff2,axiom,
! [A2: a,B2: a] :
( ( bot_bot_set_a
= ( set_or5139330845457685135Than_a @ A2 @ B2 ) )
= ( ~ ( ord_less_a @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_860_atLeastLessThan__empty__iff2,axiom,
! [A2: nat,B2: nat] :
( ( bot_bot_set_nat
= ( set_or4665077453230672383an_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_nat @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_861_greaterThan__subset__iff,axiom,
! [X4: a,Y3: a] :
( ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ X4 ) @ ( set_or8632414552788122084Than_a @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X4 ) ) ).
% greaterThan_subset_iff
thf(fact_862_greaterThanLessThan__empty,axiom,
! [L: set_a,K: set_a] :
( ( ord_less_eq_set_a @ L @ K )
=> ( ( set_or6017932776736107018_set_a @ K @ L )
= bot_bot_set_set_a ) ) ).
% greaterThanLessThan_empty
thf(fact_863_greaterThanAtMost__empty,axiom,
! [L: set_a,K: set_a] :
( ( ord_less_eq_set_a @ L @ K )
=> ( ( set_or2503527069484367278_set_a @ K @ L )
= bot_bot_set_set_a ) ) ).
% greaterThanAtMost_empty
thf(fact_864_greaterThanAtMost__empty__iff,axiom,
! [K: a,L: a] :
( ( ( set_or4472690218693186638Most_a @ K @ L )
= bot_bot_set_a )
= ( ~ ( ord_less_a @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_865_greaterThanAtMost__empty__iff,axiom,
! [K: nat,L: nat] :
( ( ( set_or6659071591806873216st_nat @ K @ L )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_866_greaterThanAtMost__empty__iff2,axiom,
! [K: a,L: a] :
( ( bot_bot_set_a
= ( set_or4472690218693186638Most_a @ K @ L ) )
= ( ~ ( ord_less_a @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_867_greaterThanAtMost__empty__iff2,axiom,
! [K: nat,L: nat] :
( ( bot_bot_set_nat
= ( set_or6659071591806873216st_nat @ K @ L ) )
= ( ~ ( ord_less_nat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_868_less__separate,axiom,
! [X4: a,Y3: a] :
( ( ord_less_a @ X4 @ Y3 )
=> ? [A3: a,B3: a] :
( ( member_a @ X4 @ ( set_ord_lessThan_a @ A3 ) )
& ( member_a @ Y3 @ ( set_or8632414552788122084Than_a @ B3 ) )
& ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ A3 ) @ ( set_or8632414552788122084Than_a @ B3 ) )
= bot_bot_set_a ) ) ) ).
% less_separate
thf(fact_869_less__separate,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ? [A3: nat,B3: nat] :
( ( member_nat @ X4 @ ( set_ord_lessThan_nat @ A3 ) )
& ( member_nat @ Y3 @ ( set_or1210151606488870762an_nat @ B3 ) )
& ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ A3 ) @ ( set_or1210151606488870762an_nat @ B3 ) )
= bot_bot_set_nat ) ) ) ).
% less_separate
thf(fact_870_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_871_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_872_equals0I,axiom,
! [A: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_873_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_874_ivl__disj__int__one_I4_J,axiom,
! [L: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(4)
thf(fact_875_ivl__disj__int__one_I7_J,axiom,
! [L: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or1269000886237332187st_nat @ L @ U ) @ ( set_or1210151606488870762an_nat @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(7)
thf(fact_876_ivl__disj__int__one_I2_J,axiom,
! [L: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(2)
thf(fact_877_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_878_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_879_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_880_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_881_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_882_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_883_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_884_not__empty__eq__Ici__eq__empty,axiom,
! [L: a] :
( bot_bot_set_a
!= ( set_ord_atLeast_a @ L ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_885_lessThan__strict__subset__iff,axiom,
! [M2: a,N2: a] :
( ( ord_less_set_a @ ( set_ord_lessThan_a @ M2 ) @ ( set_ord_lessThan_a @ N2 ) )
= ( ord_less_a @ M2 @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_886_lessThan__strict__subset__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_887_Ioi__le__Ico,axiom,
! [A2: a] : ( ord_less_eq_set_a @ ( set_or8632414552788122084Than_a @ A2 ) @ ( set_ord_atLeast_a @ A2 ) ) ).
% Ioi_le_Ico
thf(fact_888_ivl__disj__int__two_I3_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(3)
thf(fact_889_Iic__subset__Iio__iff,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A2 ) @ ( set_ord_lessThan_nat @ B2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% Iic_subset_Iio_iff
thf(fact_890_Iic__subset__Iio__iff,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ A2 ) @ ( set_ord_lessThan_a @ B2 ) )
= ( ord_less_a @ A2 @ B2 ) ) ).
% Iic_subset_Iio_iff
thf(fact_891_ivl__disj__un__one_I2_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( set_ord_lessThan_nat @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_892_Ici__subset__Ioi__iff,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ A2 ) @ ( set_or1210151606488870762an_nat @ B2 ) )
= ( ord_less_nat @ B2 @ A2 ) ) ).
% Ici_subset_Ioi_iff
thf(fact_893_Ici__subset__Ioi__iff,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ A2 ) @ ( set_or8632414552788122084Than_a @ B2 ) )
= ( ord_less_a @ B2 @ A2 ) ) ).
% Ici_subset_Ioi_iff
thf(fact_894_atLeastLessThan__def,axiom,
( set_or5139330845457685135Than_a
= ( ^ [L3: a,U2: a] : ( inf_inf_set_a @ ( set_ord_atLeast_a @ L3 ) @ ( set_ord_lessThan_a @ U2 ) ) ) ) ).
% atLeastLessThan_def
thf(fact_895_atLeastLessThan__def,axiom,
( set_or4665077453230672383an_nat
= ( ^ [L3: nat,U2: nat] : ( inf_inf_set_nat @ ( set_ord_atLeast_nat @ L3 ) @ ( set_ord_lessThan_nat @ U2 ) ) ) ) ).
% atLeastLessThan_def
thf(fact_896_ivl__disj__int__two_I7_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(7)
thf(fact_897_ivl__disj__int__two_I4_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or5834768355832116004an_nat @ M2 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(4)
thf(fact_898_ivl__disj__int__two_I5_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or5834768355832116004an_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(5)
thf(fact_899_ivl__disj__int__two_I8_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or6659071591806873216st_nat @ M2 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(8)
thf(fact_900_ivl__disj__int__two_I1_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or5834768355832116004an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(1)
thf(fact_901_ivl__disj__int__one_I8_J,axiom,
! [L: a,U: a] :
( ( inf_inf_set_a @ ( set_or5139330845457685135Than_a @ L @ U ) @ ( set_ord_atLeast_a @ U ) )
= bot_bot_set_a ) ).
% ivl_disj_int_one(8)
thf(fact_902_ivl__disj__int__one_I8_J,axiom,
! [L: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ U ) @ ( set_ord_atLeast_nat @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(8)
thf(fact_903_interval__cases,axiom,
! [S: set_nat] :
( ! [A3: nat,B3: nat,X2: nat] :
( ( member_nat @ A3 @ S )
=> ( ( member_nat @ B3 @ S )
=> ( ( ord_less_eq_nat @ A3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ B3 )
=> ( member_nat @ X2 @ S ) ) ) ) )
=> ? [A3: nat,B3: nat] :
( ( S = bot_bot_set_nat )
| ( S = top_top_set_nat )
| ( S
= ( set_ord_lessThan_nat @ B3 ) )
| ( S
= ( set_ord_atMost_nat @ B3 ) )
| ( S
= ( set_or1210151606488870762an_nat @ A3 ) )
| ( S
= ( set_ord_atLeast_nat @ A3 ) )
| ( S
= ( set_or5834768355832116004an_nat @ A3 @ B3 ) )
| ( S
= ( set_or6659071591806873216st_nat @ A3 @ B3 ) )
| ( S
= ( set_or4665077453230672383an_nat @ A3 @ B3 ) )
| ( S
= ( set_or1269000886237332187st_nat @ A3 @ B3 ) ) ) ) ).
% interval_cases
thf(fact_904_subset__emptyI,axiom,
! [A: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_905_ivl__disj__un__singleton_I4_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( set_or5939364468397584554Than_a @ L @ U ) @ ( insert_a @ U @ bot_bot_set_a ) )
= ( set_or4472690218693186638Most_a @ L @ U ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_906_ivl__disj__un__singleton_I4_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_907_ivl__disj__un__singleton_I3_J,axiom,
! [L: a,U: a] :
( ( ord_less_a @ L @ U )
=> ( ( sup_sup_set_a @ ( insert_a @ L @ bot_bot_set_a ) @ ( set_or5939364468397584554Than_a @ L @ U ) )
= ( set_or5139330845457685135Than_a @ L @ U ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_908_ivl__disj__un__singleton_I3_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_909_ivl__disj__un__singleton_I5_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_910_UNIV__I,axiom,
! [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% UNIV_I
thf(fact_911_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_912_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_913_insert__subset,axiom,
! [X4: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_914_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_915_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_916_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_917_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_918_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_919_closed__UNIV,axiom,
topolo784654279908865136osed_a @ top_top_set_a ).
% closed_UNIV
thf(fact_920_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_921_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B2: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_922_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_923_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_924_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_925_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_926_atLeastAtMost__singleton,axiom,
! [A2: nat] :
( ( set_or1269000886237332187st_nat @ A2 @ A2 )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_927_atLeastAtMost__singleton__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
= ( insert_nat @ C @ bot_bot_set_nat ) )
= ( ( A2 = B2 )
& ( B2 = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_928_closed__singleton,axiom,
! [A2: a] : ( topolo784654279908865136osed_a @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% closed_singleton
thf(fact_929_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_930_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_931_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_932_insert__subsetI,axiom,
! [X4: a,A: set_a,X6: set_a] :
( ( member_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X4 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_933_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B8: set_a] :
( ( A
= ( insert_a @ A2 @ B8 ) )
& ~ ( member_a @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_934_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_a] :
( ( A
= ( insert_a @ B2 @ C4 ) )
& ~ ( member_a @ B2 @ C4 )
& ( B
= ( insert_a @ A2 @ C4 ) )
& ~ ( member_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_935_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_936_insert__ident,axiom,
! [X4: a,A: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ~ ( member_a @ X4 @ B )
=> ( ( ( insert_a @ X4 @ A )
= ( insert_a @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_937_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_938_Set_Oset__insert,axiom,
! [X4: a,A: set_a] :
( ( member_a @ X4 @ A )
=> ~ ! [B8: set_a] :
( ( A
= ( insert_a @ X4 @ B8 ) )
=> ( member_a @ X4 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_939_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_940_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_941_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_942_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_943_insert__mono,axiom,
! [C2: set_a,D3: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D3 )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_944_subset__insert,axiom,
! [X4: a,A: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_945_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_946_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_947_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_948_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_949_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_950_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_951_closed__insert,axiom,
! [S: set_a,A2: a] :
( ( topolo784654279908865136osed_a @ S )
=> ( topolo784654279908865136osed_a @ ( insert_a @ A2 @ S ) ) ) ).
% closed_insert
thf(fact_952_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_953_not__UNIV__eq__Icc,axiom,
! [L2: nat,H2: nat] :
( top_top_set_nat
!= ( set_or1269000886237332187st_nat @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_954_subset__singletonD,axiom,
! [A: set_a,X4: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_955_subset__singleton__iff,axiom,
! [X6: set_a,A2: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_956_atLeastAtMost__singleton_H,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
=> ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_957_not__UNIV__le__Icc,axiom,
! [L: nat,H: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) ) ).
% not_UNIV_le_Icc
thf(fact_958_Iio__Int__singleton,axiom,
! [X4: a,K: a] :
( ( ( ord_less_a @ X4 @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X4 @ bot_bot_set_a ) )
= ( insert_a @ X4 @ bot_bot_set_a ) ) )
& ( ~ ( ord_less_a @ X4 @ K )
=> ( ( inf_inf_set_a @ ( set_ord_lessThan_a @ K ) @ ( insert_a @ X4 @ bot_bot_set_a ) )
= bot_bot_set_a ) ) ) ).
% Iio_Int_singleton
thf(fact_959_Iio__Int__singleton,axiom,
! [X4: nat,K: nat] :
( ( ( ord_less_nat @ X4 @ K )
=> ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X4 @ K )
=> ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ) ).
% Iio_Int_singleton
thf(fact_960_atMost__Int__atLeast,axiom,
! [N2: a] :
( ( inf_inf_set_a @ ( set_ord_atMost_a @ N2 ) @ ( set_ord_atLeast_a @ N2 ) )
= ( insert_a @ N2 @ bot_bot_set_a ) ) ).
% atMost_Int_atLeast
thf(fact_961_ivl__disj__un__singleton_I1_J,axiom,
! [L: a] :
( ( sup_sup_set_a @ ( insert_a @ L @ bot_bot_set_a ) @ ( set_or8632414552788122084Than_a @ L ) )
= ( set_ord_atLeast_a @ L ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_962_ivl__disj__un__singleton_I6_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_963_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2496817875450240012_set_a @ inf_inf_set_a @ top_top_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_964_top_Oordering__top__axioms,axiom,
ordering_top_set_a @ ord_less_eq_set_a @ ord_less_set_a @ top_top_set_a ).
% top.ordering_top_axioms
thf(fact_965_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a,Y5: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y5 @ A )
=> ( X2 = Y5 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_966_connected__closed,axiom,
( topolo2370605967727889109cted_a
= ( ^ [S2: set_a] :
~ ? [A5: set_a,B5: set_a] :
( ( topolo784654279908865136osed_a @ A5 )
& ( topolo784654279908865136osed_a @ B5 )
& ( ord_less_eq_set_a @ S2 @ ( sup_sup_set_a @ A5 @ B5 ) )
& ( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ S2 )
= bot_bot_set_a )
& ( ( inf_inf_set_a @ A5 @ S2 )
!= bot_bot_set_a )
& ( ( inf_inf_set_a @ B5 @ S2 )
!= bot_bot_set_a ) ) ) ) ).
% connected_closed
thf(fact_967_connected__closedD,axiom,
! [S3: set_a,A: set_a,B: set_a] :
( ( topolo2370605967727889109cted_a @ S3 )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ S3 )
= bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ S3 @ ( sup_sup_set_a @ A @ B ) )
=> ( ( topolo784654279908865136osed_a @ A )
=> ( ( topolo784654279908865136osed_a @ B )
=> ( ( ( inf_inf_set_a @ A @ S3 )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ B @ S3 )
= bot_bot_set_a ) ) ) ) ) ) ) ).
% connected_closedD
thf(fact_968_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
( set_or6659071591806873216st_nat
= ( ^ [A4: nat,B4: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A4 @ B4 ) @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_969_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_970_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_971_closed__Inter,axiom,
! [K2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ K2 )
=> ( topolo784654279908865136osed_a @ X2 ) )
=> ( topolo784654279908865136osed_a @ ( comple6135023378680113637_set_a @ K2 ) ) ) ).
% closed_Inter
thf(fact_972_Diff__insert0,axiom,
! [X4: a,A: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_973_insert__Diff1,axiom,
! [X4: a,B: set_a,A: set_a] :
( ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_974_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_975_Inf__atLeastAtMost,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ X4 @ Y3 ) )
= X4 ) ) ).
% Inf_atLeastAtMost
thf(fact_976_cInf__atLeastAtMost,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( comple6135023378680113637_set_a @ ( set_or6288561110385358355_set_a @ Y3 @ X4 ) )
= Y3 ) ) ).
% cInf_atLeastAtMost
thf(fact_977_cInf__atLeastAtMost,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( complete_Inf_Inf_nat @ ( set_or1269000886237332187st_nat @ Y3 @ X4 ) )
= Y3 ) ) ).
% cInf_atLeastAtMost
thf(fact_978_ivl__diff,axiom,
! [I2: nat,N2: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ M2 ) @ ( set_or4665077453230672383an_nat @ I2 @ N2 ) )
= ( set_or4665077453230672383an_nat @ N2 @ M2 ) ) ) ).
% ivl_diff
thf(fact_979_cInf__atLeastLessThan,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_nat @ Y3 @ X4 )
=> ( ( complete_Inf_Inf_nat @ ( set_or4665077453230672383an_nat @ Y3 @ X4 ) )
= Y3 ) ) ).
% cInf_atLeastLessThan
thf(fact_980_lessThan__minus__lessThan,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( set_ord_lessThan_nat @ M2 ) )
= ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ).
% lessThan_minus_lessThan
thf(fact_981_connectedD__interval,axiom,
! [U3: set_a,X4: a,Y3: a,Z4: a] :
( ( topolo2370605967727889109cted_a @ U3 )
=> ( ( member_a @ X4 @ U3 )
=> ( ( member_a @ Y3 @ U3 )
=> ( ( ord_less_eq_a @ X4 @ Z4 )
=> ( ( ord_less_eq_a @ Z4 @ Y3 )
=> ( member_a @ Z4 @ U3 ) ) ) ) ) ) ).
% connectedD_interval
thf(fact_982_cInf__eq__minimum,axiom,
! [Z4: set_a,X6: set_set_a] :
( ( member_set_a @ Z4 @ X6 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ Z4 @ X2 ) )
=> ( ( comple6135023378680113637_set_a @ X6 )
= Z4 ) ) ) ).
% cInf_eq_minimum
thf(fact_983_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_984_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_985_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_986_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_987_double__diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_988_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_989_Diff__mono,axiom,
! [A: set_a,C2: set_a,D3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ D3 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C2 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_990_insert__Diff__if,axiom,
! [X4: a,B: set_a,A: set_a] :
( ( ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( insert_a @ X4 @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_991_cInf__eq__non__empty,axiom,
! [X6: set_set_a,A2: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ A2 @ X2 ) )
=> ( ! [Y5: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ X6 )
=> ( ord_less_eq_set_a @ Y5 @ X ) )
=> ( ord_less_eq_set_a @ Y5 @ A2 ) )
=> ( ( comple6135023378680113637_set_a @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_992_cInf__greatest,axiom,
! [X6: set_set_a,Z4: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ Z4 @ X2 ) )
=> ( ord_less_eq_set_a @ Z4 @ ( comple6135023378680113637_set_a @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_993_cInf__lessD,axiom,
! [X6: set_nat,Z4: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X6 ) @ Z4 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ X6 )
& ( ord_less_nat @ X2 @ Z4 ) ) ) ) ).
% cInf_lessD
thf(fact_994_connected__contains__Icc,axiom,
! [A: set_a,A2: a,B2: a] :
( ( topolo2370605967727889109cted_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ A )
=> ( ord_less_eq_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ A ) ) ) ) ).
% connected_contains_Icc
thf(fact_995_connected__contains__Icc,axiom,
! [A: set_nat,A2: nat,B2: nat] :
( ( topolo3071422859925075001ed_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( member_nat @ B2 @ A )
=> ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ A ) ) ) ) ).
% connected_contains_Icc
thf(fact_996_connected__contains__Ioo,axiom,
! [A: set_a,A2: a,B2: a] :
( ( topolo2370605967727889109cted_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ A )
=> ( ord_less_eq_set_a @ ( set_or5939364468397584554Than_a @ A2 @ B2 ) @ A ) ) ) ) ).
% connected_contains_Ioo
thf(fact_997_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X4: a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X4 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_998_insert__Diff,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_999_Diff__insert__absorb,axiom,
! [X4: a,A: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ ( insert_a @ X4 @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1000_Diff__partition,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1001_Diff__subset__conv,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
= ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1002_subset__insert__iff,axiom,
! [A: set_a,X4: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( ( ( member_a @ X4 @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X4 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1003_Diff__single__insert,axiom,
! [A: set_a,X4: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1004_psubset__insert__iff,axiom,
! [A: set_a,X4: a,B: set_a] :
( ( ord_less_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( ( ( member_a @ X4 @ B )
=> ( ord_less_set_a @ A @ B ) )
& ( ~ ( member_a @ X4 @ B )
=> ( ( ( member_a @ X4 @ A )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X4 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1005_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or4665077453230672383an_nat
= ( ^ [A4: nat,B4: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A4 @ B4 ) @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_1006_atLeastAtMost__diff__ends,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
= ( set_or5834768355832116004an_nat @ A2 @ B2 ) ) ).
% atLeastAtMost_diff_ends
thf(fact_1007_connected__closed__set,axiom,
! [S: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo2370605967727889109cted_a @ S )
= ( ~ ? [A5: set_a,B5: set_a] :
( ( topolo784654279908865136osed_a @ A5 )
& ( topolo784654279908865136osed_a @ B5 )
& ( A5 != bot_bot_set_a )
& ( B5 != bot_bot_set_a )
& ( ( sup_sup_set_a @ A5 @ B5 )
= S )
& ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a ) ) ) ) ) ).
% connected_closed_set
thf(fact_1008_connected__as__closed__union,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( topolo2370605967727889109cted_a @ C2 )
=> ( ( C2
= ( sup_sup_set_a @ A @ B ) )
=> ( ( topolo784654279908865136osed_a @ A )
=> ( ( topolo784654279908865136osed_a @ B )
=> ( ( A != bot_bot_set_a )
=> ( ( B != bot_bot_set_a )
=> ( ( inf_inf_set_a @ A @ B )
!= bot_bot_set_a ) ) ) ) ) ) ) ).
% connected_as_closed_union
thf(fact_1009_less__eq__Inf__inter,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B ) ) ) ).
% less_eq_Inf_inter
thf(fact_1010_Inter__lower,axiom,
! [B: set_a,A: set_set_a] :
( ( member_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ B ) ) ).
% Inter_lower
thf(fact_1011_Inter__greatest,axiom,
! [A: set_set_a,C2: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A )
=> ( ord_less_eq_set_a @ C2 @ X7 ) )
=> ( ord_less_eq_set_a @ C2 @ ( comple6135023378680113637_set_a @ A ) ) ) ).
% Inter_greatest
thf(fact_1012_Inter__anti__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inter_anti_mono
thf(fact_1013_Inter__subset,axiom,
! [A: set_set_a,B: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A )
=> ( ord_less_eq_set_a @ X7 @ B ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ B ) ) ) ).
% Inter_subset
thf(fact_1014_Inf__eqI,axiom,
! [A: set_set_a,X4: set_a] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A )
=> ( ord_less_eq_set_a @ X4 @ I3 ) )
=> ( ! [Y5: set_a] :
( ! [I4: set_a] :
( ( member_set_a @ I4 @ A )
=> ( ord_less_eq_set_a @ Y5 @ I4 ) )
=> ( ord_less_eq_set_a @ Y5 @ X4 ) )
=> ( ( comple6135023378680113637_set_a @ A )
= X4 ) ) ) ).
% Inf_eqI
thf(fact_1015_Inf__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ! [B3: set_a] :
( ( member_set_a @ B3 @ B )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ X @ B3 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_mono
thf(fact_1016_Inf__lower,axiom,
! [X4: set_a,A: set_set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ X4 ) ) ).
% Inf_lower
thf(fact_1017_Inf__lower2,axiom,
! [U: set_a,A: set_set_a,V: set_a] :
( ( member_set_a @ U @ A )
=> ( ( ord_less_eq_set_a @ U @ V )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_1018_le__Inf__iff,axiom,
! [B2: set_a,A: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( comple6135023378680113637_set_a @ A ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_1019_Inf__greatest,axiom,
! [A: set_set_a,Z4: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ Z4 @ X2 ) )
=> ( ord_less_eq_set_a @ Z4 @ ( comple6135023378680113637_set_a @ A ) ) ) ).
% Inf_greatest
thf(fact_1020_Inter__Un__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B ) ) ) ).
% Inter_Un_subset
thf(fact_1021_Inf__superset__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_superset_mono
thf(fact_1022_Inf__less__eq,axiom,
! [A: set_set_a,U: set_a] :
( ! [V2: set_a] :
( ( member_set_a @ V2 @ A )
=> ( ord_less_eq_set_a @ V2 @ U ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1023_diff__shunt__var,axiom,
! [X4: set_a,Y3: set_a] :
( ( ( minus_minus_set_a @ X4 @ Y3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1024_connected__def,axiom,
( topolo2370605967727889109cted_a
= ( ^ [S4: set_a] :
~ ? [A5: set_a,B5: set_a] :
( ( topolo8477419352202985285open_a @ A5 )
& ( topolo8477419352202985285open_a @ B5 )
& ( ord_less_eq_set_a @ S4 @ ( sup_sup_set_a @ A5 @ B5 ) )
& ( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ S4 )
= bot_bot_set_a )
& ( ( inf_inf_set_a @ A5 @ S4 )
!= bot_bot_set_a )
& ( ( inf_inf_set_a @ B5 @ S4 )
!= bot_bot_set_a ) ) ) ) ).
% connected_def
thf(fact_1025_member__remove,axiom,
! [X4: a,Y3: a,A: set_a] :
( ( member_a @ X4 @ ( remove_a @ Y3 @ A ) )
= ( ( member_a @ X4 @ A )
& ( X4 != Y3 ) ) ) ).
% member_remove
thf(fact_1026_mono__atLeast,axiom,
! [B: a] : ( extend2808419353335425523_set_a @ ( set_ord_atLeast_a @ B ) ) ).
% mono_atLeast
thf(fact_1027_closed__Diff,axiom,
! [S: set_a,T3: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( topolo8477419352202985285open_a @ T3 )
=> ( topolo784654279908865136osed_a @ ( minus_minus_set_a @ S @ T3 ) ) ) ) ).
% closed_Diff
thf(fact_1028_open__Diff,axiom,
! [S: set_a,T3: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( topolo8477419352202985285open_a @ ( minus_minus_set_a @ S @ T3 ) ) ) ) ).
% open_Diff
thf(fact_1029_open__subopen,axiom,
( topolo8477419352202985285open_a
= ( ^ [S4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ S4 )
=> ? [T4: set_a] :
( ( topolo8477419352202985285open_a @ T4 )
& ( member_a @ X3 @ T4 )
& ( ord_less_eq_set_a @ T4 @ S4 ) ) ) ) ) ).
% open_subopen
thf(fact_1030_topological__space__class_OopenI,axiom,
! [S: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ S )
=> ? [T5: set_a] :
( ( topolo8477419352202985285open_a @ T5 )
& ( member_a @ X2 @ T5 )
& ( ord_less_eq_set_a @ T5 @ S ) ) )
=> ( topolo8477419352202985285open_a @ S ) ) ).
% topological_space_class.openI
thf(fact_1031_t0__space,axiom,
! [X4: a,Y3: a] :
( ( X4 != Y3 )
=> ? [U4: set_a] :
( ( topolo8477419352202985285open_a @ U4 )
& ( ( member_a @ X4 @ U4 )
!= ( member_a @ Y3 @ U4 ) ) ) ) ).
% t0_space
thf(fact_1032_t1__space,axiom,
! [X4: a,Y3: a] :
( ( X4 != Y3 )
=> ? [U4: set_a] :
( ( topolo8477419352202985285open_a @ U4 )
& ( member_a @ X4 @ U4 )
& ~ ( member_a @ Y3 @ U4 ) ) ) ).
% t1_space
thf(fact_1033_separation__t0,axiom,
! [X4: a,Y3: a] :
( ( X4 != Y3 )
= ( ? [U5: set_a] :
( ( topolo8477419352202985285open_a @ U5 )
& ( ( member_a @ X4 @ U5 )
!= ( member_a @ Y3 @ U5 ) ) ) ) ) ).
% separation_t0
thf(fact_1034_separation__t1,axiom,
! [X4: a,Y3: a] :
( ( X4 != Y3 )
= ( ? [U5: set_a] :
( ( topolo8477419352202985285open_a @ U5 )
& ( member_a @ X4 @ U5 )
& ~ ( member_a @ Y3 @ U5 ) ) ) ) ).
% separation_t1
thf(fact_1035_hausdorff,axiom,
! [X4: a,Y3: a] :
( ( X4 != Y3 )
=> ? [U4: set_a,V3: set_a] :
( ( topolo8477419352202985285open_a @ U4 )
& ( topolo8477419352202985285open_a @ V3 )
& ( member_a @ X4 @ U4 )
& ( member_a @ Y3 @ V3 )
& ( ( inf_inf_set_a @ U4 @ V3 )
= bot_bot_set_a ) ) ) ).
% hausdorff
thf(fact_1036_separation__t2,axiom,
! [X4: a,Y3: a] :
( ( X4 != Y3 )
= ( ? [U5: set_a,V4: set_a] :
( ( topolo8477419352202985285open_a @ U5 )
& ( topolo8477419352202985285open_a @ V4 )
& ( member_a @ X4 @ U5 )
& ( member_a @ Y3 @ V4 )
& ( ( inf_inf_set_a @ U5 @ V4 )
= bot_bot_set_a ) ) ) ) ).
% separation_t2
thf(fact_1037_mono__set,axiom,
( extend2808419353335425523_set_a
= ( ^ [S4: set_a] :
! [X3: a,Y2: a] :
( ( ord_less_eq_a @ X3 @ Y2 )
=> ( ( member_a @ X3 @ S4 )
=> ( member_a @ Y2 @ S4 ) ) ) ) ) ).
% mono_set
thf(fact_1038_mono__set,axiom,
( extend347329919781519187_set_a
= ( ^ [S4: set_set_a] :
! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ( member_set_a @ X3 @ S4 )
=> ( member_set_a @ Y2 @ S4 ) ) ) ) ) ).
% mono_set
thf(fact_1039_open__right,axiom,
! [S: set_a,X4: a,Y3: a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( member_a @ X4 @ S )
=> ( ( ord_less_a @ X4 @ Y3 )
=> ? [B3: a] :
( ( ord_less_a @ X4 @ B3 )
& ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ X4 @ B3 ) @ S ) ) ) ) ) ).
% open_right
thf(fact_1040_open__right,axiom,
! [S: set_nat,X4: nat,Y3: nat] :
( ( topolo4328251076210115529en_nat @ S )
=> ( ( member_nat @ X4 @ S )
=> ( ( ord_less_nat @ X4 @ Y3 )
=> ? [B3: nat] :
( ( ord_less_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ X4 @ B3 ) @ S ) ) ) ) ) ).
% open_right
thf(fact_1041_open__left,axiom,
! [S: set_nat,X4: nat,Y3: nat] :
( ( topolo4328251076210115529en_nat @ S )
=> ( ( member_nat @ X4 @ S )
=> ( ( ord_less_nat @ Y3 @ X4 )
=> ? [B3: nat] :
( ( ord_less_nat @ B3 @ X4 )
& ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ B3 @ X4 ) @ S ) ) ) ) ) ).
% open_left
thf(fact_1042_open__left,axiom,
! [S: set_a,X4: a,Y3: a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( member_a @ X4 @ S )
=> ( ( ord_less_a @ Y3 @ X4 )
=> ? [B3: a] :
( ( ord_less_a @ B3 @ X4 )
& ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ B3 @ X4 ) @ S ) ) ) ) ) ).
% open_left
thf(fact_1043_connected__diff__open__from__closed,axiom,
! [S3: set_a,T: set_a,U: set_a] :
( ( ord_less_eq_set_a @ S3 @ T )
=> ( ( ord_less_eq_set_a @ T @ U )
=> ( ( topolo8477419352202985285open_a @ S3 )
=> ( ( topolo784654279908865136osed_a @ T )
=> ( ( topolo2370605967727889109cted_a @ U )
=> ( ( topolo2370605967727889109cted_a @ ( minus_minus_set_a @ T @ S3 ) )
=> ( topolo2370605967727889109cted_a @ ( minus_minus_set_a @ U @ S3 ) ) ) ) ) ) ) ) ).
% connected_diff_open_from_closed
thf(fact_1044_connectedD,axiom,
! [A: set_a,U3: set_a,V5: set_a] :
( ( topolo2370605967727889109cted_a @ A )
=> ( ( topolo8477419352202985285open_a @ U3 )
=> ( ( topolo8477419352202985285open_a @ V5 )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ U3 @ V5 ) @ A )
= bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ U3 @ V5 ) )
=> ( ( ( inf_inf_set_a @ U3 @ A )
= bot_bot_set_a )
| ( ( inf_inf_set_a @ V5 @ A )
= bot_bot_set_a ) ) ) ) ) ) ) ).
% connectedD
thf(fact_1045_connectedI,axiom,
! [U3: set_a] :
( ! [A8: set_a] :
( ( topolo8477419352202985285open_a @ A8 )
=> ! [B8: set_a] :
( ( topolo8477419352202985285open_a @ B8 )
=> ( ( ( inf_inf_set_a @ A8 @ U3 )
!= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ B8 @ U3 )
!= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( inf_inf_set_a @ A8 @ B8 ) @ U3 )
= bot_bot_set_a )
=> ~ ( ord_less_eq_set_a @ U3 @ ( sup_sup_set_a @ A8 @ B8 ) ) ) ) ) ) )
=> ( topolo2370605967727889109cted_a @ U3 ) ) ).
% connectedI
thf(fact_1046_at__within__nhd,axiom,
! [X4: a,S: set_a,T3: set_a,U3: set_a] :
( ( member_a @ X4 @ S )
=> ( ( topolo8477419352202985285open_a @ S )
=> ( ( ( minus_minus_set_a @ ( inf_inf_set_a @ T3 @ S ) @ ( insert_a @ X4 @ bot_bot_set_a ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ U3 @ S ) @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
=> ( ( topolo1902352237885396414thin_a @ X4 @ T3 )
= ( topolo1902352237885396414thin_a @ X4 @ U3 ) ) ) ) ) ).
% at_within_nhd
thf(fact_1047_at__le,axiom,
! [S3: set_a,T: set_a,X4: a] :
( ( ord_less_eq_set_a @ S3 @ T )
=> ( ord_less_eq_filter_a @ ( topolo1902352237885396414thin_a @ X4 @ S3 ) @ ( topolo1902352237885396414thin_a @ X4 @ T ) ) ) ).
% at_le
thf(fact_1048_at__within__open,axiom,
! [A2: a,S: set_a] :
( ( member_a @ A2 @ S )
=> ( ( topolo8477419352202985285open_a @ S )
=> ( ( topolo1902352237885396414thin_a @ A2 @ S )
= ( topolo1902352237885396414thin_a @ A2 @ top_top_set_a ) ) ) ) ).
% at_within_open
thf(fact_1049_at__within__Icc__at,axiom,
! [A2: a,X4: a,B2: a] :
( ( ord_less_a @ A2 @ X4 )
=> ( ( ord_less_a @ X4 @ B2 )
=> ( ( topolo1902352237885396414thin_a @ X4 @ ( set_or672772299803893939Most_a @ A2 @ B2 ) )
= ( topolo1902352237885396414thin_a @ X4 @ top_top_set_a ) ) ) ) ).
% at_within_Icc_at
thf(fact_1050_at__within__Icc__at,axiom,
! [A2: nat,X4: nat,B2: nat] :
( ( ord_less_nat @ A2 @ X4 )
=> ( ( ord_less_nat @ X4 @ B2 )
=> ( ( topolo4659099751122792720in_nat @ X4 @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
= ( topolo4659099751122792720in_nat @ X4 @ top_top_set_nat ) ) ) ) ).
% at_within_Icc_at
thf(fact_1051_at__within__open__subset,axiom,
! [A2: a,S: set_a,T3: set_a] :
( ( member_a @ A2 @ S )
=> ( ( topolo8477419352202985285open_a @ S )
=> ( ( ord_less_eq_set_a @ S @ T3 )
=> ( ( topolo1902352237885396414thin_a @ A2 @ T3 )
= ( topolo1902352237885396414thin_a @ A2 @ top_top_set_a ) ) ) ) ) ).
% at_within_open_subset
thf(fact_1052_at__within__Icc__at__left,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( topolo1902352237885396414thin_a @ B2 @ ( set_or672772299803893939Most_a @ A2 @ B2 ) )
= ( topolo1902352237885396414thin_a @ B2 @ ( set_ord_lessThan_a @ B2 ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_1053_at__within__Icc__at__left,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( topolo4659099751122792720in_nat @ B2 @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
= ( topolo4659099751122792720in_nat @ B2 @ ( set_ord_lessThan_nat @ B2 ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_1054_at__within__Icc__at__right,axiom,
! [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
=> ( ( topolo1902352237885396414thin_a @ A2 @ ( set_or672772299803893939Most_a @ A2 @ B2 ) )
= ( topolo1902352237885396414thin_a @ A2 @ ( set_or8632414552788122084Than_a @ A2 ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_1055_at__within__Icc__at__right,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( topolo4659099751122792720in_nat @ A2 @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
= ( topolo4659099751122792720in_nat @ A2 @ ( set_or1210151606488870762an_nat @ A2 ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_1056_less__eq__cInf__inter,axiom,
! [A: set_set_a,B: set_set_a] :
( ( condit8937546108433946286_set_a @ A )
=> ( ( condit8937546108433946286_set_a @ B )
=> ( ( ( inf_inf_set_set_a @ A @ B )
!= bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1057_chains__extend,axiom,
! [C: set_set_a,S: set_set_a,Z4: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ Z4 @ S )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ C )
=> ( ord_less_eq_set_a @ X2 @ Z4 ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z4 @ bot_bot_set_set_a ) @ C ) @ ( chains_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1058_bdd__below_OI,axiom,
! [A: set_a,M3: a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_a @ M3 @ X2 ) )
=> ( condit5901475214736682318elow_a @ A ) ) ).
% bdd_below.I
thf(fact_1059_bdd__below_OI,axiom,
! [A: set_set_a,M3: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M3 @ X2 ) )
=> ( condit8937546108433946286_set_a @ A ) ) ).
% bdd_below.I
thf(fact_1060_bdd__belowI,axiom,
! [A: set_a,M2: a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_a @ M2 @ X2 ) )
=> ( condit5901475214736682318elow_a @ A ) ) ).
% bdd_belowI
thf(fact_1061_bdd__belowI,axiom,
! [A: set_set_a,M2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M2 @ X2 ) )
=> ( condit8937546108433946286_set_a @ A ) ) ).
% bdd_belowI
thf(fact_1062_bdd__below__Icc,axiom,
! [A2: nat,B2: nat] : ( condit1738341127787009408ow_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) ) ).
% bdd_below_Icc
thf(fact_1063_bdd__below__Ico,axiom,
! [A2: nat,B2: nat] : ( condit1738341127787009408ow_nat @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) ) ).
% bdd_below_Ico
thf(fact_1064_bdd__below__Ici,axiom,
! [A2: a] : ( condit5901475214736682318elow_a @ ( set_ord_atLeast_a @ A2 ) ) ).
% bdd_below_Ici
thf(fact_1065_compact__Int__closed,axiom,
! [S: set_a,T3: set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( topolo8439159285038550427pact_a @ ( inf_inf_set_a @ S @ T3 ) ) ) ) ).
% compact_Int_closed
thf(fact_1066_bdd__below_Ounfold,axiom,
( condit8937546108433946286_set_a
= ( ^ [A5: set_set_a] :
? [M4: set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( ord_less_eq_set_a @ M4 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1067_bdd__below_OE,axiom,
! [A: set_a] :
( ( condit5901475214736682318elow_a @ A )
=> ~ ! [M5: a] :
~ ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_a @ M5 @ X ) ) ) ).
% bdd_below.E
thf(fact_1068_bdd__below_OE,axiom,
! [A: set_set_a] :
( ( condit8937546108433946286_set_a @ A )
=> ~ ! [M5: set_a] :
~ ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ M5 @ X ) ) ) ).
% bdd_below.E
thf(fact_1069_bdd__below__mono,axiom,
! [B: set_a,A: set_a] :
( ( condit5901475214736682318elow_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( condit5901475214736682318elow_a @ A ) ) ) ).
% bdd_below_mono
thf(fact_1070_cInf__lower2,axiom,
! [X4: set_a,X6: set_set_a,Y3: set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( condit8937546108433946286_set_a @ X6 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X6 ) @ Y3 ) ) ) ) ).
% cInf_lower2
thf(fact_1071_cInf__lower,axiom,
! [X4: set_a,X6: set_set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ( condit8937546108433946286_set_a @ X6 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X6 ) @ X4 ) ) ) ).
% cInf_lower
thf(fact_1072_compact__imp__closed,axiom,
! [S3: set_a] :
( ( topolo8439159285038550427pact_a @ S3 )
=> ( topolo784654279908865136osed_a @ S3 ) ) ).
% compact_imp_closed
thf(fact_1073_Zorn__Lemma2,axiom,
! [A: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( chains_a @ A ) )
=> ? [Xa: set_a] :
( ( member_set_a @ Xa @ A )
& ! [Xb: set_a] :
( ( member_set_a @ Xb @ X2 )
=> ( ord_less_eq_set_a @ Xb @ Xa ) ) ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1074_chainsD,axiom,
! [C: set_set_a,S: set_set_a,X4: set_a,Y3: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ X4 @ C )
=> ( ( member_set_a @ Y3 @ C )
=> ( ( ord_less_eq_set_a @ X4 @ Y3 )
| ( ord_less_eq_set_a @ Y3 @ X4 ) ) ) ) ) ).
% chainsD
thf(fact_1075_le__cInf__iff,axiom,
! [S: set_set_a,A2: set_a] :
( ( S != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ S )
=> ( ( ord_less_eq_set_a @ A2 @ ( comple6135023378680113637_set_a @ S ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ S )
=> ( ord_less_eq_set_a @ A2 @ X3 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1076_cInf__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( B != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ A )
=> ( ! [B3: set_a] :
( ( member_set_a @ B3 @ B )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ X @ B3 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B ) ) ) ) ) ).
% cInf_mono
thf(fact_1077_cInf__less__iff,axiom,
! [X6: set_nat,Y3: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ X6 )
=> ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X6 ) @ Y3 )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ X6 )
& ( ord_less_nat @ X3 @ Y3 ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_1078_cInf__superset__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ( condit8937546108433946286_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ B ) @ ( comple6135023378680113637_set_a @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1079_closed__Int__compact,axiom,
! [S3: set_a,T: set_a] :
( ( topolo784654279908865136osed_a @ S3 )
=> ( ( topolo8439159285038550427pact_a @ T )
=> ( topolo8439159285038550427pact_a @ ( inf_inf_set_a @ S3 @ T ) ) ) ) ).
% closed_Int_compact
thf(fact_1080_cInf__le__iff,axiom,
! [A: set_nat,X4: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ X4 )
= ( ! [Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% cInf_le_iff
thf(fact_1081_compact__imp__fip,axiom,
! [S: set_a,F2: set_set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ! [T6: set_a] :
( ( member_set_a @ T6 @ F2 )
=> ( topolo784654279908865136osed_a @ T6 ) )
=> ( ! [F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ F2 )
=> ( ( inf_inf_set_a @ S @ ( comple6135023378680113637_set_a @ F3 ) )
!= bot_bot_set_a ) ) )
=> ( ( inf_inf_set_a @ S @ ( comple6135023378680113637_set_a @ F2 ) )
!= bot_bot_set_a ) ) ) ) ).
% compact_imp_fip
thf(fact_1082_image__eqI,axiom,
! [B2: nat,F: nat > nat,X4: nat,A: set_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1083_image__eqI,axiom,
! [B2: a,F: a > a,X4: a,A: set_a] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1084_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1085_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1086_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1087_image__insert,axiom,
! [F: nat > nat,A2: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_1088_insert__image,axiom,
! [X4: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X4 @ A )
=> ( ( insert_nat @ ( F @ X4 ) @ ( image_nat_nat @ F @ A ) )
= ( image_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_1089_range__eqI,axiom,
! [B2: nat,F: nat > nat,X4: nat] :
( ( B2
= ( F @ X4 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1090_rangeI,axiom,
! [F: nat > nat,X4: nat] : ( member_nat @ ( F @ X4 ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1091_finite__imp__closed,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( topolo784654279908865136osed_a @ S ) ) ).
% finite_imp_closed
thf(fact_1092_imageI,axiom,
! [X4: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_1093_imageI,axiom,
! [X4: a,A: set_a,F: a > a] :
( ( member_a @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_1094_image__iff,axiom,
! [Z4: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z4 @ ( image_nat_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z4
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_1095_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
& ( P @ X ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_1096_image__cong,axiom,
! [M3: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
( ( M3 = N3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M3 )
= ( image_nat_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_1097_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X: nat] :
( ( member_nat @ X @ A )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_1098_rev__image__eqI,axiom,
! [X4: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1099_rev__image__eqI,axiom,
! [X4: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1100_finite__imp__less__Inf,axiom,
! [X6: set_nat,X4: nat,A2: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X4 @ X6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_nat @ A2 @ X2 ) )
=> ( ord_less_nat @ A2 @ ( complete_Inf_Inf_nat @ X6 ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_1101_cInf__le__finite,axiom,
! [X6: set_set_a,X4: set_a] :
( ( finite_finite_set_a @ X6 )
=> ( ( member_set_a @ X4 @ X6 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ X6 ) @ X4 ) ) ) ).
% cInf_le_finite
thf(fact_1102_image__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_1103_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_1104_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_1105_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1106_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1107_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
=> ( B
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1108_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A )
=> ( B
!= ( image_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1109_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_1110_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1111_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1112_bdd__belowI2,axiom,
! [A: set_nat,M2: nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ M2 @ ( F @ X2 ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1113_bdd__belowI2,axiom,
! [A: set_a,M2: set_a,F: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M2 @ ( F @ X2 ) ) )
=> ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1114_bdd__below_OI2,axiom,
! [A: set_nat,M3: nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ M3 @ ( F @ X2 ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1115_bdd__below_OI2,axiom,
! [A: set_a,M3: set_a,F: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M3 @ ( F @ X2 ) ) )
=> ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1116_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1117_image__Int__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1118_INF__eq,axiom,
! [A: set_a,B: set_a,G: a > set_a,F: a > set_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X: a] :
( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ ( G @ X ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ J2 ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) )
= ( comple6135023378680113637_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1119_range__subsetD,axiom,
! [F: nat > nat,B: set_nat,I2: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B )
=> ( member_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_1120_the__elem__image__unique,axiom,
! [A: set_nat,F: nat > nat,X4: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ A )
=> ( ( F @ Y5 )
= ( F @ X4 ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A ) )
= ( F @ X4 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1121_range__eq__singletonD,axiom,
! [F: nat > nat,A2: nat,X4: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F @ X4 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_1122_INF__eq__iff,axiom,
! [I5: set_a,F: a > set_a,C: set_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_less_eq_set_a @ ( F @ I3 ) @ C ) )
=> ( ( ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ I5 ) )
= C )
= ( ! [X3: a] :
( ( member_a @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1123_cINF__greatest,axiom,
! [A: set_nat,M2: nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ M2 @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ M2 @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1124_cINF__greatest,axiom,
! [A: set_a,M2: set_a,F: a > set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ M2 @ ( F @ X2 ) ) )
=> ( ord_less_eq_set_a @ M2 @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_1125_in__image__insert__iff,axiom,
! [B: set_set_a,X4: a,A: set_a] :
( ! [C5: set_a] :
( ( member_set_a @ C5 @ B )
=> ~ ( member_a @ X4 @ C5 ) )
=> ( ( member_set_a @ A @ ( image_set_a_set_a @ ( insert_a @ X4 ) @ B ) )
= ( ( member_a @ X4 @ A )
& ( member_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1126_cINF__lower2,axiom,
! [F: nat > nat,A: set_nat,X4: nat,U: nat] :
( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( member_nat @ X4 @ A )
=> ( ( ord_less_eq_nat @ ( F @ X4 ) @ U )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_1127_cINF__lower2,axiom,
! [F: a > set_a,A: set_a,X4: a,U: set_a] :
( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) )
=> ( ( member_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ ( F @ X4 ) @ U )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_1128_cINF__lower,axiom,
! [F: nat > nat,A: set_nat,X4: nat] :
( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) @ ( F @ X4 ) ) ) ) ).
% cINF_lower
thf(fact_1129_cINF__lower,axiom,
! [F: a > set_a,A: set_a,X4: a] :
( ( condit8937546108433946286_set_a @ ( image_a_set_a @ F @ A ) )
=> ( ( member_a @ X4 @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) ) @ ( F @ X4 ) ) ) ) ).
% cINF_lower
thf(fact_1130_finite__less__Inf__iff,axiom,
! [X6: set_nat,A2: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ A2 @ ( complete_Inf_Inf_nat @ X6 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_nat @ A2 @ X3 ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_1131_cINF__mono,axiom,
! [B: set_nat,F: nat > nat,A: set_nat,G: nat > nat] :
( ( B != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ! [M6: nat] :
( ( member_nat @ M6 @ B )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ ( F @ X ) @ ( G @ M6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ G @ B ) ) ) ) ) ) ).
% cINF_mono
thf(fact_1132_cINF__mono,axiom,
! [B: set_a,F: nat > nat,A: set_nat,G: a > nat] :
( ( B != bot_bot_set_a )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ! [M6: a] :
( ( member_a @ M6 @ B )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ ( F @ X ) @ ( G @ M6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_a_nat @ G @ B ) ) ) ) ) ) ).
% cINF_mono
thf(fact_1133_le__cINF__iff,axiom,
! [A: set_nat,F: nat > nat,U: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ U @ ( F @ X3 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_1134_cINF__le__iff,axiom,
! [A: set_nat,F: nat > nat,X4: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) @ X4 )
= ( ! [Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ) ) ) ).
% cINF_le_iff
thf(fact_1135_cINF__superset__mono,axiom,
! [A: set_nat,G: nat > nat,B: set_nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ G @ B ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ G @ B ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1136_cINF__superset__mono,axiom,
! [A: set_a,G: a > set_a,B: set_a,F: a > set_a] :
( ( A != bot_bot_set_a )
=> ( ( condit8937546108433946286_set_a @ ( image_a_set_a @ G @ B ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ord_less_eq_set_a @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G @ B ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1137_cINF__insert,axiom,
! [A: set_nat,F: nat > nat,A2: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ ( insert_nat @ A2 @ A ) ) )
= ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_1138_cINF__union,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( B != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ B ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ B ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_1139_compact__fip,axiom,
( topolo8439159285038550427pact_a
= ( ^ [U5: set_a] :
! [A5: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( topolo784654279908865136osed_a @ X3 ) )
=> ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
=> ( ( finite_finite_set_a @ B5 )
=> ( ( inf_inf_set_a @ U5 @ ( comple6135023378680113637_set_a @ B5 ) )
!= bot_bot_set_a ) ) )
=> ( ( inf_inf_set_a @ U5 @ ( comple6135023378680113637_set_a @ A5 ) )
!= bot_bot_set_a ) ) ) ) ) ).
% compact_fip
thf(fact_1140_finite__has__minimal2,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ord_less_eq_a @ X2 @ A2 )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1141_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1142_finite__has__maximal2,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ord_less_eq_a @ A2 @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1143_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A2 @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1144_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1145_all__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1146_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_1147_infinite__super,axiom,
! [S: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S @ T3 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T3 ) ) ) ).
% infinite_super
thf(fact_1148_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_1149_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1150_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1151_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1152_all__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A ) )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1153_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A )
& ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1154_ex__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A )
& ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1155_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
& ( finite_finite_nat @ C5 )
& ( B
= ( image_nat_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1156_finite__subset__image,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A )
& ( finite_finite_a @ C5 )
& ( B
= ( image_a_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1157_finite__surj,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1158_finite__subset__induct,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A3 @ A )
=> ( ~ ( member_a @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A3 @ F4 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1159_finite__subset__induct_H,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A3 @ A )
=> ( ( ord_less_eq_set_a @ F4 @ A )
=> ( ~ ( member_a @ A3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A3 @ F4 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1160_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1161_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1162_finite__linorder__min__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [B3: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ! [X: a] :
( ( member_a @ X @ A8 )
=> ( ord_less_a @ B3 @ X ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_a @ B3 @ A8 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1163_finite__linorder__min__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A8 )
=> ( ord_less_nat @ B3 @ X ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B3 @ A8 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1164_finite__linorder__max__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [B3: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ! [X: a] :
( ( member_a @ X @ A8 )
=> ( ord_less_a @ X @ B3 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_a @ B3 @ A8 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1165_finite__linorder__max__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A8 )
=> ( ord_less_nat @ X @ B3 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B3 @ A8 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1166_infinite__growing,axiom,
! [X6: set_a] :
( ( X6 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ X6 )
=> ? [Xa: a] :
( ( member_a @ Xa @ X6 )
& ( ord_less_a @ X2 @ Xa ) ) )
=> ~ ( finite_finite_a @ X6 ) ) ) ).
% infinite_growing
thf(fact_1167_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X6 )
& ( ord_less_nat @ X2 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_1168_ex__min__if__finite,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ S )
& ~ ? [Xa: a] :
( ( member_a @ Xa @ S )
& ( ord_less_a @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1169_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1170_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_1171_Fpow__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A ) @ ( finite_Fpow_a @ B ) ) ) ).
% Fpow_mono
thf(fact_1172_closure__closed,axiom,
! [S: set_a] :
( ( topolo784654279908865136osed_a @ S )
=> ( ( elementary_closure_a @ S )
= S ) ) ).
% closure_closed
thf(fact_1173_closed__closure,axiom,
! [S: set_a] : ( topolo784654279908865136osed_a @ ( elementary_closure_a @ S ) ) ).
% closed_closure
thf(fact_1174_closure__subset,axiom,
! [S: set_a] : ( ord_less_eq_set_a @ S @ ( elementary_closure_a @ S ) ) ).
% closure_subset
thf(fact_1175_closure__mono,axiom,
! [S: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S @ T3 )
=> ( ord_less_eq_set_a @ ( elementary_closure_a @ S ) @ ( elementary_closure_a @ T3 ) ) ) ).
% closure_mono
thf(fact_1176_closure__subset__eq,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ ( elementary_closure_a @ S ) @ S )
= ( topolo784654279908865136osed_a @ S ) ) ).
% closure_subset_eq
thf(fact_1177_closure__minimal,axiom,
! [S: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S @ T3 )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( ord_less_eq_set_a @ ( elementary_closure_a @ S ) @ T3 ) ) ) ).
% closure_minimal
thf(fact_1178_closure__unique,axiom,
! [S: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S @ T3 )
=> ( ( topolo784654279908865136osed_a @ T3 )
=> ( ! [T7: set_a] :
( ( ord_less_eq_set_a @ S @ T7 )
=> ( ( topolo784654279908865136osed_a @ T7 )
=> ( ord_less_eq_set_a @ T3 @ T7 ) ) )
=> ( ( elementary_closure_a @ S )
= T3 ) ) ) ) ).
% closure_unique
thf(fact_1179_closure__eq,axiom,
! [S: set_a] :
( ( ( elementary_closure_a @ S )
= S )
= ( topolo784654279908865136osed_a @ S ) ) ).
% closure_eq
thf(fact_1180_closure__open__Int__superset,axiom,
! [S: set_a,T3: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( elementary_closure_a @ T3 ) )
=> ( ( elementary_closure_a @ ( inf_inf_set_a @ S @ T3 ) )
= ( elementary_closure_a @ S ) ) ) ) ).
% closure_open_Int_superset
thf(fact_1181_open__Int__closure__subset,axiom,
! [S: set_a,T3: set_a] :
( ( topolo8477419352202985285open_a @ S )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ S @ ( elementary_closure_a @ T3 ) ) @ ( elementary_closure_a @ ( inf_inf_set_a @ S @ T3 ) ) ) ) ).
% open_Int_closure_subset
thf(fact_1182_closure__iff__nhds__not__empty,axiom,
! [X4: a,X6: set_a] :
( ( member_a @ X4 @ ( elementary_closure_a @ X6 ) )
= ( ! [A5: set_a,S4: set_a] :
( ( ord_less_eq_set_a @ S4 @ A5 )
=> ( ( topolo8477419352202985285open_a @ S4 )
=> ( ( member_a @ X4 @ S4 )
=> ( ( inf_inf_set_a @ X6 @ A5 )
!= bot_bot_set_a ) ) ) ) ) ) ).
% closure_iff_nhds_not_empty
thf(fact_1183_connected__intermediate__closure,axiom,
! [S3: set_a,T: set_a] :
( ( topolo2370605967727889109cted_a @ S3 )
=> ( ( ord_less_eq_set_a @ S3 @ T )
=> ( ( ord_less_eq_set_a @ T @ ( elementary_closure_a @ S3 ) )
=> ( topolo2370605967727889109cted_a @ T ) ) ) ) ).
% connected_intermediate_closure
thf(fact_1184_compact__eq__Heine__Borel,axiom,
( topolo8439159285038550427pact_a
= ( ^ [S4: set_a] :
! [C4: set_set_a] :
( ( ! [X3: set_a] :
( ( member_set_a @ X3 @ C4 )
=> ( topolo8477419352202985285open_a @ X3 ) )
& ( ord_less_eq_set_a @ S4 @ ( comple2307003609928055243_set_a @ C4 ) ) )
=> ? [D4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ D4 @ C4 )
& ( finite_finite_set_a @ D4 )
& ( ord_less_eq_set_a @ S4 @ ( comple2307003609928055243_set_a @ D4 ) ) ) ) ) ) ).
% compact_eq_Heine_Borel
thf(fact_1185_cSup__atLeastAtMost,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( comple2307003609928055243_set_a @ ( set_or6288561110385358355_set_a @ Y3 @ X4 ) )
= X4 ) ) ).
% cSup_atLeastAtMost
thf(fact_1186_cSup__atLeastAtMost,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( complete_Sup_Sup_nat @ ( set_or1269000886237332187st_nat @ Y3 @ X4 ) )
= X4 ) ) ).
% cSup_atLeastAtMost
thf(fact_1187_Sup__atLeastAtMost,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( comple2307003609928055243_set_a @ ( set_or6288561110385358355_set_a @ X4 @ Y3 ) )
= Y3 ) ) ).
% Sup_atLeastAtMost
thf(fact_1188_cSup__greaterThanAtMost,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_nat @ Y3 @ X4 )
=> ( ( complete_Sup_Sup_nat @ ( set_or6659071591806873216st_nat @ Y3 @ X4 ) )
= X4 ) ) ).
% cSup_greaterThanAtMost
thf(fact_1189_closed__Union,axiom,
! [S: set_set_a] :
( ( finite_finite_set_a @ S )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ S )
=> ( topolo784654279908865136osed_a @ X2 ) )
=> ( topolo784654279908865136osed_a @ ( comple2307003609928055243_set_a @ S ) ) ) ) ).
% closed_Union
thf(fact_1190_Union__least,axiom,
! [A: set_set_a,C2: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A )
=> ( ord_less_eq_set_a @ X7 @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_1191_Union__upper,axiom,
! [B: set_a,A: set_set_a] :
( ( member_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Union_upper
thf(fact_1192_Union__subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ? [Y: set_a] :
( ( member_set_a @ Y @ B )
& ( ord_less_eq_set_a @ X2 @ Y ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_subsetI
thf(fact_1193_Sup__eqI,axiom,
! [A: set_set_a,X4: set_a] :
( ! [Y5: set_a] :
( ( member_set_a @ Y5 @ A )
=> ( ord_less_eq_set_a @ Y5 @ X4 ) )
=> ( ! [Y5: set_a] :
( ! [Z2: set_a] :
( ( member_set_a @ Z2 @ A )
=> ( ord_less_eq_set_a @ Z2 @ Y5 ) )
=> ( ord_less_eq_set_a @ X4 @ Y5 ) )
=> ( ( comple2307003609928055243_set_a @ A )
= X4 ) ) ) ).
% Sup_eqI
thf(fact_1194_Sup__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [A3: set_a] :
( ( member_set_a @ A3 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ B )
& ( ord_less_eq_set_a @ A3 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_mono
thf(fact_1195_Sup__least,axiom,
! [A: set_set_a,Z4: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ Z4 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ Z4 ) ) ).
% Sup_least
thf(fact_1196_Sup__upper,axiom,
! [X4: set_a,A: set_set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ X4 @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Sup_upper
thf(fact_1197_Sup__le__iff,axiom,
! [A: set_set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ B2 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ X3 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_1198_Sup__upper2,axiom,
! [U: set_a,A: set_set_a,V: set_a] :
( ( member_set_a @ U @ A )
=> ( ( ord_less_eq_set_a @ V @ U )
=> ( ord_less_eq_set_a @ V @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1199_cSup__eq__maximum,axiom,
! [Z4: set_a,X6: set_set_a] :
( ( member_set_a @ Z4 @ X6 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ X2 @ Z4 ) )
=> ( ( comple2307003609928055243_set_a @ X6 )
= Z4 ) ) ) ).
% cSup_eq_maximum
thf(fact_1200_less__cSupE,axiom,
! [Y3: nat,X6: set_nat] :
( ( ord_less_nat @ Y3 @ ( complete_Sup_Sup_nat @ X6 ) )
=> ( ( X6 != bot_bot_set_nat )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ) ) ).
% less_cSupE
thf(fact_1201_less__cSupD,axiom,
! [X6: set_nat,Z4: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z4 @ ( complete_Sup_Sup_nat @ X6 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ X6 )
& ( ord_less_nat @ Z4 @ X2 ) ) ) ) ).
% less_cSupD
thf(fact_1202_cSup__eq__non__empty,axiom,
! [X6: set_set_a,A2: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ X2 @ A2 ) )
=> ( ! [Y5: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ X6 )
=> ( ord_less_eq_set_a @ X @ Y5 ) )
=> ( ord_less_eq_set_a @ A2 @ Y5 ) )
=> ( ( comple2307003609928055243_set_a @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1203_cSup__least,axiom,
! [X6: set_set_a,Z4: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ X2 @ Z4 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ X6 ) @ Z4 ) ) ) ).
% cSup_least
thf(fact_1204_less__eq__Sup,axiom,
! [A: set_set_a,U: set_a] :
( ! [V2: set_a] :
( ( member_set_a @ V2 @ A )
=> ( ord_less_eq_set_a @ U @ V2 ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1205_Sup__subset__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_1206_Union__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_mono
thf(fact_1207_Zorn__Lemma,axiom,
! [A: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( chains_a @ A ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ X2 ) @ A ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_1208_SUP__eq,axiom,
! [A: set_a,B: set_a,F: a > set_a,G: a > set_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X: a] :
( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ ( F @ I3 ) @ ( G @ X ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( ord_less_eq_set_a @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1209_le__cSup__finite,axiom,
! [X6: set_set_a,X4: set_a] :
( ( finite_finite_set_a @ X6 )
=> ( ( member_set_a @ X4 @ X6 )
=> ( ord_less_eq_set_a @ X4 @ ( comple2307003609928055243_set_a @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_1210_finite__imp__Sup__less,axiom,
! [X6: set_nat,X4: nat,A2: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X4 @ X6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_nat @ X2 @ A2 ) )
=> ( ord_less_nat @ ( complete_Sup_Sup_nat @ X6 ) @ A2 ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_1211_SUP__eq__iff,axiom,
! [I5: set_a,C: set_a,F: a > set_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_less_eq_set_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ I5 ) )
= C )
= ( ! [X3: a] :
( ( member_a @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1212_cSUP__least,axiom,
! [A: set_nat,F: nat > nat,M3: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1213_cSUP__least,axiom,
! [A: set_a,F: a > set_a,M3: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ M3 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1214_finite__Sup__less__iff,axiom,
! [X6: set_nat,A2: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X6 ) @ A2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_nat @ X3 @ A2 ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_1215_Sup__inter__less__eq,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1216_Inf__le__Sup,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Inf_le_Sup
thf(fact_1217_Union__Int__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_Int_subset
thf(fact_1218_compactE,axiom,
! [S: set_a,T8: set_set_a] :
( ( topolo8439159285038550427pact_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( comple2307003609928055243_set_a @ T8 ) )
=> ( ! [B8: set_a] :
( ( member_set_a @ B8 @ T8 )
=> ( topolo8477419352202985285open_a @ B8 ) )
=> ~ ! [T9: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T9 @ T8 )
=> ( ( finite_finite_set_a @ T9 )
=> ~ ( ord_less_eq_set_a @ S @ ( comple2307003609928055243_set_a @ T9 ) ) ) ) ) ) ) ).
% compactE
thf(fact_1219_compactI,axiom,
! [S3: set_a] :
( ! [C5: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ C5 )
=> ( topolo8477419352202985285open_a @ X ) )
=> ( ( ord_less_eq_set_a @ S3 @ ( comple2307003609928055243_set_a @ C5 ) )
=> ? [C6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C6 @ C5 )
& ( finite_finite_set_a @ C6 )
& ( ord_less_eq_set_a @ S3 @ ( comple2307003609928055243_set_a @ C6 ) ) ) ) )
=> ( topolo8439159285038550427pact_a @ S3 ) ) ).
% compactI
thf(fact_1220_finite__subset__Union,axiom,
! [A: set_a,B9: set_set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( comple2307003609928055243_set_a @ B9 ) )
=> ~ ! [F5: set_set_a] :
( ( finite_finite_set_a @ F5 )
=> ( ( ord_le3724670747650509150_set_a @ F5 @ B9 )
=> ~ ( ord_less_eq_set_a @ A @ ( comple2307003609928055243_set_a @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1221_finite__subset__Union__chain,axiom,
! [A: set_a,B9: set_set_a,A9: set_set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( comple2307003609928055243_set_a @ B9 ) )
=> ( ( B9 != bot_bot_set_set_a )
=> ( ( pred_chain_set_a @ A9 @ ord_less_set_a @ B9 )
=> ~ ! [B8: set_a] :
( ( member_set_a @ B8 @ B9 )
=> ~ ( ord_less_eq_set_a @ A @ B8 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_1222_cInf__le__cSup,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ A )
=> ( ( condit8937546108433946286_set_a @ A )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple2307003609928055243_set_a @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_1223_bdd__above_OI,axiom,
! [A: set_a,M3: a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_a @ X2 @ M3 ) )
=> ( condit5209368051240477026bove_a @ A ) ) ).
% bdd_above.I
thf(fact_1224_bdd__above_OI,axiom,
! [A: set_set_a,M3: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ M3 ) )
=> ( condit3373647341569784514_set_a @ A ) ) ).
% bdd_above.I
thf(fact_1225_bdd__above__Icc,axiom,
! [A2: nat,B2: nat] : ( condit2214826472909112428ve_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) ) ).
% bdd_above_Icc
thf(fact_1226_bdd__above__Ico,axiom,
! [A2: nat,B2: nat] : ( condit2214826472909112428ve_nat @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) ) ).
% bdd_above_Ico
thf(fact_1227_less__cSup__iff,axiom,
! [X6: set_nat,Y3: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X6 )
=> ( ( ord_less_nat @ Y3 @ ( complete_Sup_Sup_nat @ X6 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ X6 )
& ( ord_less_nat @ Y3 @ X3 ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_1228_cSup__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( B != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ A )
=> ( ! [B3: set_a] :
( ( member_set_a @ B3 @ B )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ B3 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ B ) @ ( comple2307003609928055243_set_a @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1229_cSup__le__iff,axiom,
! [S: set_set_a,A2: set_a] :
( ( S != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ S )
=> ( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ S ) @ A2 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ S )
=> ( ord_less_eq_set_a @ X3 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1230_cSup__upper2,axiom,
! [X4: set_a,X6: set_set_a,Y3: set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( condit3373647341569784514_set_a @ X6 )
=> ( ord_less_eq_set_a @ Y3 @ ( comple2307003609928055243_set_a @ X6 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1231_cSup__upper,axiom,
! [X4: set_a,X6: set_set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ( condit3373647341569784514_set_a @ X6 )
=> ( ord_less_eq_set_a @ X4 @ ( comple2307003609928055243_set_a @ X6 ) ) ) ) ).
% cSup_upper
thf(fact_1232_cSUP__upper2,axiom,
! [F: nat > nat,A: set_nat,X4: nat,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( member_nat @ X4 @ A )
=> ( ( ord_less_eq_nat @ U @ ( F @ X4 ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1233_cSUP__upper2,axiom,
! [F: a > set_a,A: set_a,X4: a,U: set_a] :
( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A ) )
=> ( ( member_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ U @ ( F @ X4 ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1234_cSUP__upper,axiom,
! [X4: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X4 @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1235_cSUP__upper,axiom,
! [X4: a,A: set_a,F: a > set_a] :
( ( member_a @ X4 @ A )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A ) )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1236_subset__Zorn_H,axiom,
! [A: set_set_a] :
( ! [C5: set_set_a] :
( ( pred_chain_set_a @ A @ ord_less_set_a @ C5 )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ C5 ) @ A ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% subset_Zorn'
thf(fact_1237_subset__chain__def,axiom,
! [A9: set_set_a,C7: set_set_a] :
( ( pred_chain_set_a @ A9 @ ord_less_set_a @ C7 )
= ( ( ord_le3724670747650509150_set_a @ C7 @ A9 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ C7 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ C7 )
=> ( ( ord_less_eq_set_a @ X3 @ Y2 )
| ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_1238_subset__chain__insert,axiom,
! [A9: set_set_a,B: set_a,B9: set_set_a] :
( ( pred_chain_set_a @ A9 @ ord_less_set_a @ ( insert_set_a @ B @ B9 ) )
= ( ( member_set_a @ B @ A9 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ B9 )
=> ( ( ord_less_eq_set_a @ X3 @ B )
| ( ord_less_eq_set_a @ B @ X3 ) ) )
& ( pred_chain_set_a @ A9 @ ord_less_set_a @ B9 ) ) ) ).
% subset_chain_insert
thf(fact_1239_subset__Zorn,axiom,
! [A: set_set_a] :
( ! [C5: set_set_a] :
( ( pred_chain_set_a @ A @ ord_less_set_a @ C5 )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ C5 )
=> ( ord_less_eq_set_a @ Xa2 @ X ) ) ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% subset_Zorn
thf(fact_1240_pred__on_Ochain__def,axiom,
( pred_chain_a
= ( ^ [A5: set_a,P4: a > a > $o,C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A5 )
& ! [X3: a] :
( ( member_a @ X3 @ C4 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ C4 )
=> ( ( sup_sup_a_a_o @ P4
@ ^ [Y4: a,Z5: a] : ( Y4 = Z5 )
@ X3
@ Y2 )
| ( sup_sup_a_a_o @ P4
@ ^ [Y4: a,Z5: a] : ( Y4 = Z5 )
@ Y2
@ X3 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_1241_pred__on_OchainI,axiom,
! [C2: set_a,A: set_a,P: a > a > $o] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ! [X2: a,Y5: a] :
( ( member_a @ X2 @ C2 )
=> ( ( member_a @ Y5 @ C2 )
=> ( ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z5: a] : ( Y4 = Z5 )
@ X2
@ Y5 )
| ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z5: a] : ( Y4 = Z5 )
@ Y5
@ X2 ) ) ) )
=> ( pred_chain_a @ A @ P @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_1242_bdd__above_Ounfold,axiom,
( condit3373647341569784514_set_a
= ( ^ [A5: set_set_a] :
? [M4: set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( ord_less_eq_set_a @ X3 @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1243_bdd__above_OE,axiom,
! [A: set_a] :
( ( condit5209368051240477026bove_a @ A )
=> ~ ! [M5: a] :
~ ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_a @ X @ M5 ) ) ) ).
% bdd_above.E
thf(fact_1244_bdd__above_OE,axiom,
! [A: set_set_a] :
( ( condit3373647341569784514_set_a @ A )
=> ~ ! [M5: set_a] :
~ ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ M5 ) ) ) ).
% bdd_above.E
thf(fact_1245_bdd__above__mono,axiom,
! [B: set_a,A: set_a] :
( ( condit5209368051240477026bove_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( condit5209368051240477026bove_a @ A ) ) ) ).
% bdd_above_mono
thf(fact_1246_pred__on_Ochain__total,axiom,
! [A: set_a,P: a > a > $o,C2: set_a,X4: a,Y3: a] :
( ( pred_chain_a @ A @ P @ C2 )
=> ( ( member_a @ X4 @ C2 )
=> ( ( member_a @ Y3 @ C2 )
=> ( ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z5: a] : ( Y4 = Z5 )
@ X4
@ Y3 )
| ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z5: a] : ( Y4 = Z5 )
@ Y3
@ X4 ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_1247_chain__mono,axiom,
! [A: set_a,P: a > a > $o,Q: a > a > $o,C2: set_a] :
( ! [X2: a,Y5: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y5 @ A )
=> ( ( P @ X2 @ Y5 )
=> ( Q @ X2 @ Y5 ) ) ) )
=> ( ( pred_chain_a @ A @ P @ C2 )
=> ( pred_chain_a @ A @ Q @ C2 ) ) ) ).
% chain_mono
thf(fact_1248_bdd__above_OI2,axiom,
! [A: set_nat,F: nat > nat,M3: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M3 ) )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1249_bdd__above_OI2,axiom,
! [A: set_a,F: a > set_a,M3: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ M3 ) )
=> ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1250_pred__on_Ochain__extend,axiom,
! [A: set_a,P: a > a > $o,C2: set_a,Z4: a] :
( ( pred_chain_a @ A @ P @ C2 )
=> ( ( member_a @ Z4 @ A )
=> ( ! [X2: a] :
( ( member_a @ X2 @ C2 )
=> ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z5: a] : ( Y4 = Z5 )
@ X2
@ Z4 ) )
=> ( pred_chain_a @ A @ P @ ( sup_sup_set_a @ ( insert_a @ Z4 @ bot_bot_set_a ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1251_le__cSup__iff,axiom,
! [A: set_nat,X4: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ( ord_less_eq_nat @ X4 @ ( complete_Sup_Sup_nat @ A ) )
= ( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ Y2 @ X3 ) ) ) ) ) ) ) ).
% le_cSup_iff
thf(fact_1252_cSUP__mono,axiom,
! [A: set_nat,G: nat > nat,B: set_nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B ) )
=> ( ! [N4: nat] :
( ( member_nat @ N4 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1253_cSUP__mono,axiom,
! [A: set_a,G: nat > nat,B: set_nat,F: a > nat] :
( ( A != bot_bot_set_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B ) )
=> ( ! [N4: a] :
( ( member_a @ N4 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1254_cSUP__le__iff,axiom,
! [A: set_nat,F: nat > nat,U: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ U )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1255_cSup__subset__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1256_not__in__connected__cases,axiom,
! [S: set_a,X4: a] :
( ( topolo2370605967727889109cted_a @ S )
=> ( ~ ( member_a @ X4 @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( ( condit5209368051240477026bove_a @ S )
=> ~ ! [Y: a] :
( ( member_a @ Y @ S )
=> ( ord_less_eq_a @ Y @ X4 ) ) )
=> ~ ( ( condit5901475214736682318elow_a @ S )
=> ~ ! [Y: a] :
( ( member_a @ Y @ S )
=> ( ord_less_eq_a @ X4 @ Y ) ) ) ) ) ) ) ).
% not_in_connected_cases
thf(fact_1257_subset__Zorn__nonempty,axiom,
! [A9: set_set_a] :
( ( A9 != bot_bot_set_set_a )
=> ( ! [C8: set_set_a] :
( ( C8 != bot_bot_set_set_a )
=> ( ( pred_chain_set_a @ A9 @ ord_less_set_a @ C8 )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ C8 ) @ A9 ) ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A9 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A9 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_1258_le__cSUP__iff,axiom,
! [A: set_nat,F: nat > nat,X4: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ X4 @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) )
= ( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ Y2 @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% le_cSUP_iff
thf(fact_1259_cSUP__subset__mono,axiom,
! [A: set_nat,G: nat > nat,B: set_nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1260_cSUP__subset__mono,axiom,
! [A: set_a,G: a > set_a,B: set_a,F: a > set_a] :
( ( A != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ G @ B ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1261_cSup__inter__less__eq,axiom,
! [A: set_set_a,B: set_set_a] :
( ( condit3373647341569784514_set_a @ A )
=> ( ( condit3373647341569784514_set_a @ B )
=> ( ( ( inf_inf_set_set_a @ A @ B )
!= bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A @ B ) ) @ ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1262_cSUP__insert,axiom,
! [A: set_nat,F: nat > nat,A2: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ ( insert_nat @ A2 @ A ) ) )
= ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_1263_cSUP__union,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( B != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ B ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ B ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1264_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic6009151579333465974et_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1265_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic8986249270076014136_set_a @ inf_inf_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1266_image__Suc__atLeastLessThan,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_1267_image__Suc__atLeastAtMost,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_1268_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1269_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1270_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1271_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1272_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
% Conjectures (1)
thf(conj_0,conjecture,
$false ).
%------------------------------------------------------------------------------