TPTP Problem File: SLH0088^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc3/0010_Prover/prob_00024_001063__11949170_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1592 ( 394 unt; 316 typ; 0 def)
% Number of atoms : 3964 (1315 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10716 ( 183 ~; 20 |; 160 &;8616 @)
% ( 0 <=>;1737 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 1701 (1701 >; 0 *; 0 +; 0 <<)
% Number of symbols : 292 ( 289 usr; 18 con; 0-4 aty)
% Number of variables : 3357 ( 117 ^;3084 !; 156 ?;3357 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:27:37.576
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J_J,type,
set_set_set_rule: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Syntax__Orule_J_J,type,
stream_stream_rule: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
set_set_set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Rat__Orat_J_J,type,
stream_stream_rat: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Nat__Onat_J_J,type,
stream_stream_nat: $tType ).
thf(ty_n_t__Stream__Ostream_It__Set__Oset_It__Syntax__Orule_J_J,type,
stream_set_rule: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Syntax__Orule_J_J,type,
set_stream_rule: $tType ).
thf(ty_n_t__Stream__Ostream_It__Set__Oset_It__Rat__Orat_J_J,type,
stream_set_rat: $tType ).
thf(ty_n_t__Stream__Ostream_It__Set__Oset_It__Nat__Onat_J_J,type,
stream_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Rat__Orat_J_J,type,
set_stream_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Nat__Onat_J_J,type,
set_stream_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
set_set_rule: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Syntax__Orule_M_Eo_J_J,type,
set_rule_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Rat__Orat_M_Eo_J_J,type,
set_rat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Stream__Ostream_It__Syntax__Orule_J,type,
stream_rule: $tType ).
thf(ty_n_t__Stream__Ostream_It__Rat__Orat_J,type,
stream_rat: $tType ).
thf(ty_n_t__Stream__Ostream_It__Nat__Onat_J,type,
stream_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Syntax__Orule_J,type,
set_rule: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Syntax__Orule,type,
rule: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (289)
thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Ofenum_001t__Syntax__Orule,type,
abstra7284221463285775110m_rule: stream_rule > stream_rule ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_Eo_J,type,
comple6214475593288795910_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Rat__Orat_M_Eo_J,type,
comple2477142665972227838_rat_o: set_rat_o > rat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Syntax__Orule_M_Eo_J,type,
comple715424409190658129rule_o: set_rule_o > rule > $o ).
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_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Rat__Orat_J,type,
comple4298007329820168263et_rat: set_set_rat > set_rat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple1065008630642458357et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
comple3908394330019556605et_rat: set_set_set_rat > set_set_rat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
comple1145120538619166314t_rule: set_set_set_rule > set_set_rule ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Syntax__Orule_J,type,
comple5773694076043965236t_rule: set_set_rule > set_rule ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Rat__Orat_M_Eo_J,type,
comple4580332206425622756_rat_o: set_rat_o > rat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Syntax__Orule_M_Eo_J,type,
comple1826244231481717815rule_o: set_rule_o > rule > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Rat__Orat_J,type,
comple3890839924845867745et_rat: set_set_rat > set_rat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
comple3392050375588816791et_rat: set_set_set_rat > set_set_rat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
comple4235951075906658052t_rule: set_set_set_rule > set_set_rule ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Syntax__Orule_J,type,
comple2146307154184993742t_rule: set_set_rule > set_rule ).
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__Rat__Orat,type,
condit1579696412822616692ve_rat: set_rat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Nat__Onat_J,type,
condit5477540289124974626et_nat: set_set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Rat__Orat_J,type,
condit1969311730731577898et_rat: set_set_rat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Syntax__Orule_J,type,
condit9083804844586111511t_rule: set_set_rule > $o ).
thf(sy_c_Countable_Ofrom__nat_001t__Nat__Onat,type,
from_nat_nat: nat > nat ).
thf(sy_c_Countable_Ofrom__nat_001t__Rat__Orat,type,
from_nat_rat: nat > rat ).
thf(sy_c_Countable_Onat__to__rat__surj,type,
nat_to_rat_surj: nat > rat ).
thf(sy_c_Countable_Oto__nat_001t__Nat__Onat,type,
to_nat_nat: nat > nat ).
thf(sy_c_Countable_Oto__nat_001t__Rat__Orat,type,
to_nat_rat: rat > nat ).
thf(sy_c_Encoding_Orule__of__nat,type,
rule_of_nat: nat > rule ).
thf(sy_c_Fair__Stream_Ofair_001t__Nat__Onat,type,
fair_fair_nat: stream_nat > $o ).
thf(sy_c_Fair__Stream_Ofair_001t__Rat__Orat,type,
fair_fair_rat: stream_rat > $o ).
thf(sy_c_Fair__Stream_Ofair_001t__Set__Oset_It__Nat__Onat_J,type,
fair_fair_set_nat: stream_set_nat > $o ).
thf(sy_c_Fair__Stream_Ofair_001t__Set__Oset_It__Rat__Orat_J,type,
fair_fair_set_rat: stream_set_rat > $o ).
thf(sy_c_Fair__Stream_Ofair_001t__Set__Oset_It__Syntax__Orule_J,type,
fair_fair_set_rule: stream_set_rule > $o ).
thf(sy_c_Fair__Stream_Ofair_001t__Syntax__Orule,type,
fair_fair_rule: stream_rule > $o ).
thf(sy_c_Fair__Stream_Ofair__stream_001t__Nat__Onat,type,
fair_fair_stream_nat: ( nat > nat ) > stream_nat ).
thf(sy_c_Fair__Stream_Ofair__stream_001t__Rat__Orat,type,
fair_fair_stream_rat: ( nat > rat ) > stream_rat ).
thf(sy_c_Fair__Stream_Ofair__stream_001t__Set__Oset_It__Nat__Onat_J,type,
fair_f5488550982935056091et_nat: ( nat > set_nat ) > stream_set_nat ).
thf(sy_c_Fair__Stream_Ofair__stream_001t__Set__Oset_It__Rat__Orat_J,type,
fair_f1980322424541659363et_rat: ( nat > set_rat ) > stream_set_rat ).
thf(sy_c_Fair__Stream_Ofair__stream_001t__Set__Oset_It__Syntax__Orule_J,type,
fair_f1814642869706786768t_rule: ( nat > set_rule ) > stream_set_rule ).
thf(sy_c_Fair__Stream_Ofair__stream_001t__Syntax__Orule,type,
fair_f4564919574533178778m_rule: ( nat > rule ) > stream_rule ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Rat__Orat,type,
finite_Fpow_rat: set_rat > set_set_rat ).
thf(sy_c_Finite__Set_OFpow_001t__Syntax__Orule,type,
finite_Fpow_rule: set_rule > set_set_rule ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Rat__Orat,type,
bij_betw_nat_rat: ( nat > rat ) > set_nat > set_rat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Syntax__Orule,type,
bij_betw_nat_rule: ( nat > rule ) > set_nat > set_rule > $o ).
thf(sy_c_Fun_Obij__betw_001t__Rat__Orat_001t__Nat__Onat,type,
bij_betw_rat_nat: ( rat > nat ) > set_rat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Rat__Orat_001t__Rat__Orat,type,
bij_betw_rat_rat: ( rat > rat ) > set_rat > set_rat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Rat__Orat_001t__Syntax__Orule,type,
bij_betw_rat_rule: ( rat > rule ) > set_rat > set_rule > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
bij_be3438014552859920132et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J,type,
bij_be9153158031321299212et_rat: ( set_nat > set_rat ) > set_set_nat > set_set_rat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Syntax__Orule_J,type,
bij_be6872371095225530105t_rule: ( set_nat > set_rule ) > set_set_nat > set_set_rule > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
bij_be8683898457559657236et_rat: ( set_rat > set_rat ) > set_set_rat > set_set_rat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Rat__Orat_J_001t__Syntax__Orule,type,
bij_be2386713868094972619t_rule: ( set_rat > rule ) > set_set_rat > set_rule > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Syntax__Orule_J_001t__Set__Oset_It__Syntax__Orule_J,type,
bij_be2582266366990741742t_rule: ( set_rule > set_rule ) > set_set_rule > set_set_rule > $o ).
thf(sy_c_Fun_Obij__betw_001t__Syntax__Orule_001t__Nat__Onat,type,
bij_betw_rule_nat: ( rule > nat ) > set_rule > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Syntax__Orule_001t__Rat__Orat,type,
bij_betw_rule_rat: ( rule > rat ) > set_rule > set_rat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Syntax__Orule_001t__Set__Oset_It__Rat__Orat_J,type,
bij_be5592134826634264011et_rat: ( rule > set_rat ) > set_rule > set_set_rat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Syntax__Orule_001t__Syntax__Orule,type,
bij_betw_rule_rule: ( rule > rule ) > set_rule > set_rule > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Rat__Orat,type,
inj_on_nat_rat: ( nat > rat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Syntax__Orule,type,
inj_on_nat_rule: ( nat > rule ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Nat__Onat,type,
inj_on_rat_nat: ( rat > nat ) > set_rat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Rat__Orat,type,
inj_on_rat_rat: ( rat > rat ) > set_rat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Syntax__Orule,type,
inj_on_rat_rule: ( rat > rule ) > set_rat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J,type,
inj_on1096178645466186887et_rat: ( set_nat > set_rat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Syntax__Orule_J,type,
inj_on4755138273128556404t_rule: ( set_nat > set_rule ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Rat__Orat_J_001t__Rat__Orat,type,
inj_on_set_rat_rat: ( set_rat > rat ) > set_set_rat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
inj_on626919071704544911et_rat: ( set_rat > set_rat ) > set_set_rat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Rat__Orat_J_001t__Syntax__Orule,type,
inj_on_set_rat_rule: ( set_rat > rule ) > set_set_rat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Syntax__Orule_J_001t__Set__Oset_It__Syntax__Orule_J,type,
inj_on733622247606601321t_rule: ( set_rule > set_rule ) > set_set_rule > $o ).
thf(sy_c_Fun_Oinj__on_001t__Syntax__Orule_001t__Nat__Onat,type,
inj_on_rule_nat: ( rule > nat ) > set_rule > $o ).
thf(sy_c_Fun_Oinj__on_001t__Syntax__Orule_001t__Rat__Orat,type,
inj_on_rule_rat: ( rule > rat ) > set_rule > $o ).
thf(sy_c_Fun_Oinj__on_001t__Syntax__Orule_001t__Set__Oset_It__Rat__Orat_J,type,
inj_on_rule_set_rat: ( rule > set_rat ) > set_rule > $o ).
thf(sy_c_Fun_Oinj__on_001t__Syntax__Orule_001t__Syntax__Orule,type,
inj_on_rule_rule: ( rule > rule ) > set_rule > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Rat__Orat,type,
monotone_on_nat_rat: set_nat > ( nat > nat > $o ) > ( rat > rat > $o ) > ( nat > rat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Rat__Orat_001t__Nat__Onat,type,
monotone_on_rat_nat: set_rat > ( rat > rat > $o ) > ( nat > nat > $o ) > ( rat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Rat__Orat_001t__Rat__Orat,type,
monotone_on_rat_rat: set_rat > ( rat > rat > $o ) > ( rat > rat > $o ) > ( rat > rat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
monoto2923694778811248831at_nat: set_set_nat > ( set_nat > set_nat > $o ) > ( nat > nat > $o ) > ( set_nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
monoto1748750089227133045et_nat: set_set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > ( set_nat > set_nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Rat__Orat_J_001t__Nat__Onat,type,
monoto7746382625157022407at_nat: set_set_rat > ( set_rat > set_rat > $o ) > ( nat > nat > $o ) > ( set_rat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
monoto6994633993926870149et_rat: set_set_rat > ( set_rat > set_rat > $o ) > ( set_rat > set_rat > $o ) > ( set_rat > set_rat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Syntax__Orule_J_001t__Nat__Onat,type,
monoto593345095951663796le_nat: set_set_rule > ( set_rule > set_rule > $o ) > ( nat > nat > $o ) > ( set_rule > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Syntax__Orule_J_001t__Set__Oset_It__Syntax__Orule_J,type,
monoto7176200889899739231t_rule: set_set_rule > ( set_rule > set_rule > $o ) > ( set_rule > set_rule > $o ) > ( set_rule > set_rule ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Syntax__Orule_001t__Nat__Onat,type,
monotone_on_rule_nat: set_rule > ( rule > rule > $o ) > ( nat > nat > $o ) > ( rule > nat ) > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Rat__Orat,type,
the_inv_into_nat_rat: set_nat > ( nat > rat ) > rat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Syntax__Orule,type,
the_in5544616208814386890t_rule: set_nat > ( nat > rule ) > rule > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Rat__Orat_001t__Nat__Onat,type,
the_inv_into_rat_nat: set_rat > ( rat > nat ) > nat > rat ).
thf(sy_c_Fun_Othe__inv__into_001t__Rat__Orat_001t__Rat__Orat,type,
the_inv_into_rat_rat: set_rat > ( rat > rat ) > rat > rat ).
thf(sy_c_Fun_Othe__inv__into_001t__Rat__Orat_001t__Syntax__Orule,type,
the_in2217467737105579218t_rule: set_rat > ( rat > rule ) > rule > rat ).
thf(sy_c_Fun_Othe__inv__into_001t__Syntax__Orule_001t__Nat__Onat,type,
the_in8781880623634246602le_nat: set_rule > ( rule > nat ) > nat > rule ).
thf(sy_c_Fun_Othe__inv__into_001t__Syntax__Orule_001t__Rat__Orat,type,
the_in8146750563547750866le_rat: set_rule > ( rule > rat ) > rat > rule ).
thf(sy_c_Fun_Othe__inv__into_001t__Syntax__Orule_001t__Syntax__Orule,type,
the_in80044576880915775e_rule: set_rule > ( rule > rule ) > rule > rule ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
minus_minus_rat: rat > rat > rat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Rat__Orat_J,type,
minus_minus_set_rat: set_rat > set_rat > set_rat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
minus_5007325069933123869et_rat: set_set_rat > set_set_rat > set_set_rat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
minus_5074010835065784970t_rule: set_set_rule > set_set_rule > set_set_rule ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Syntax__Orule_J,type,
minus_minus_set_rule: set_rule > set_rule > set_rule ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
uminus_uminus_rat: rat > rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Rat__Orat_J,type,
uminus2201863774496077783et_rat: set_rat > set_rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
uminus613421341184616069et_nat: set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
uminus3456807040561714317et_rat: set_set_rat > set_set_rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
uminus4253708974368314874t_rule: set_set_rule > set_set_rule ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Syntax__Orule_J,type,
uminus4869265918275750596t_rule: set_rule > set_rule ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Nat__Onat,type,
hilber5277034221543178913on_nat: ( nat > nat ) > $o ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Rat__Orat,type,
hilber4641904161456683177on_rat: ( rat > rat ) > $o ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Set__Oset_It__Nat__Onat_J,type,
hilber6749216848459082327et_nat: ( set_nat > set_nat ) > $o ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Set__Oset_It__Rat__Orat_J,type,
hilber3240988290065685599et_rat: ( set_rat > set_rat ) > $o ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Set__Oset_It__Syntax__Orule_J,type,
hilber5661084989754287948t_rule: ( set_rule > set_rule ) > $o ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Syntax__Orule,type,
hilber6733072011887318294n_rule: ( rule > rule ) > $o ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Nat__Onat,type,
hilber3633877196798814958at_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Rat__Orat,type,
hilber2998747136712319222at_rat: set_nat > ( nat > rat ) > rat > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Syntax__Orule,type,
hilber8541579349336805475t_rule: set_nat > ( nat > rule ) > rule > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Rat__Orat_001t__Nat__Onat,type,
hilber3317322552863949046at_nat: set_rat > ( rat > nat ) > nat > rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Rat__Orat_001t__Rat__Orat,type,
hilber2682192492777453310at_rat: set_rat > ( rat > rat ) > rat > rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Rat__Orat_001t__Syntax__Orule,type,
hilber5214430877627997803t_rule: set_rat > ( rat > rule ) > rule > rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
hilber7439080120785757786et_nat: set_set_nat > ( set_nat > set_nat ) > set_nat > set_nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__Rat__Orat_J_001t__Rat__Orat,type,
hilber7959523950909280948at_rat: set_set_rat > ( set_rat > rat ) > rat > set_rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
hilber3461591988630719082et_rat: set_set_rat > ( set_rat > set_rat ) > set_rat > set_rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__Rat__Orat_J_001t__Syntax__Orule,type,
hilber3948050505403073569t_rule: set_set_rat > ( set_rat > rule ) > rule > set_rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__Syntax__Orule_J_001t__Set__Oset_It__Syntax__Orule_J,type,
hilber824017158074262596t_rule: set_set_rule > ( set_rule > set_rule ) > set_rule > set_rule ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Syntax__Orule_001t__Nat__Onat,type,
hilber2555471727301889379le_nat: set_rule > ( rule > nat ) > nat > rule ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Syntax__Orule_001t__Rat__Orat,type,
hilber1920341667215393643le_rat: set_rule > ( rule > rat ) > rat > rule ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Syntax__Orule_001t__Set__Oset_It__Rat__Orat_J,type,
hilber7153471463942364961et_rat: set_rule > ( rule > set_rat ) > set_rat > rule ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Syntax__Orule_001t__Syntax__Orule,type,
hilber2978553400015838680e_rule: set_rule > ( rule > rule ) > rule > rule ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_It__Nat__Onat_J,type,
comple1596078789208929544et_nat: ( set_nat > set_nat ) > set_nat ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_It__Rat__Orat_J,type,
comple7311222267670308624et_rat: ( set_rat > set_rat ) > set_rat ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_It__Syntax__Orule_J,type,
comple2988633178464755965t_rule: ( set_rule > set_rule ) > set_rule ).
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__Rat__Orat_J,type,
bot_bot_set_rat: set_rat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
bot_bot_set_set_rule: set_set_rule ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Syntax__Orule_J,type,
bot_bot_set_rule: set_rule ).
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__Rat__Orat,type,
ord_less_rat: rat > rat > $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__Rat__Orat_J,type,
ord_less_set_rat: set_rat > set_rat > $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__Rat__Orat_J_J,type,
ord_less_set_set_rat: set_set_rat > set_set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
ord_le3472952955546049285t_rule: set_set_rule > set_set_rule > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Syntax__Orule_J,type,
ord_less_set_rule: set_rule > set_rule > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $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__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $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__Rat__Orat_J_J,type,
ord_le513522071413781156et_rat: set_set_rat > set_set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
ord_le7968974978423766289t_rule: set_set_rule > set_set_rule > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Syntax__Orule_J,type,
ord_less_eq_set_rule: set_rule > set_rule > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Rat__Orat_J,type,
ordering_top_set_rat: ( set_rat > set_rat > $o ) > ( set_rat > set_rat > $o ) > set_rat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
orderi1027199981026551883et_nat: ( set_set_nat > set_set_nat > $o ) > ( set_set_nat > set_set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
orderi3870585680403650131et_rat: ( set_set_rat > set_set_rat > $o ) > ( set_set_rat > set_set_rat > $o ) > set_set_rat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
orderi4207796640154259904t_rule: ( set_set_rule > set_set_rule > $o ) > ( set_set_rule > set_set_rule > $o ) > set_set_rule > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Syntax__Orule_J,type,
orderi2038897200410189450t_rule: ( set_rule > set_rule > $o ) > ( set_rule > set_rule > $o ) > set_rule > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Rat__Orat_M_Eo_J,type,
top_top_rat_o: rat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
top_top_set_nat_o: set_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
top_top_set_rat_o: set_rat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_It__Syntax__Orule_J_M_Eo_J,type,
top_top_set_rule_o: set_rule > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Syntax__Orule_M_Eo_J,type,
top_top_rule_o: rule > $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_It__Rat__Orat_J,type,
top_top_set_rat: set_rat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
top_top_set_set_rat: set_set_rat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
top_to1114752748106383522et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
top_to535976598521457322et_rat: set_set_set_rat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J_J,type,
top_to1078124695766848407t_rule: set_set_set_rule ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
top_top_set_set_rule: set_set_rule ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Syntax__Orule_J,type,
top_top_set_rule: set_rule ).
thf(sy_c_Prover_Orules,type,
rules: stream_rule ).
thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Rat__Orat,type,
field_6020823756834552118ts_rat: set_rat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
collect_rat: ( rat > $o ) > set_rat ).
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__Rat__Orat_J,type,
collect_set_rat: ( set_rat > $o ) > set_set_rat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Syntax__Orule_J,type,
collect_set_rule: ( set_rule > $o ) > set_set_rule ).
thf(sy_c_Set_OCollect_001t__Syntax__Orule,type,
collect_rule: ( rule > $o ) > set_rule ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Rat__Orat,type,
pow_rat: set_rat > set_set_rat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Nat__Onat_J,type,
pow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Rat__Orat_J,type,
pow_set_rat: set_set_rat > set_set_set_rat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Syntax__Orule_J,type,
pow_set_rule: set_set_rule > set_set_set_rule ).
thf(sy_c_Set_OPow_001t__Syntax__Orule,type,
pow_rule: set_rule > set_set_rule ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Rat__Orat_M_Eo_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_rat_o_set_rat: ( ( rat > $o ) > set_rat ) > set_rat_o > set_set_rat ).
thf(sy_c_Set_Oimage_001_062_It__Syntax__Orule_M_Eo_J_001t__Set__Oset_It__Syntax__Orule_J,type,
image_1281159361656534528t_rule: ( ( rule > $o ) > set_rule ) > set_rule_o > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Rat__Orat,type,
image_nat_rat: ( nat > rat ) > set_nat > set_rat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Rat__Orat_J,type,
image_nat_set_rat: ( nat > set_rat ) > set_nat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Syntax__Orule_J,type,
image_nat_set_rule: ( nat > set_rule ) > set_nat > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Syntax__Orule,type,
image_nat_rule: ( nat > rule ) > set_nat > set_rule ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Nat__Onat,type,
image_rat_nat: ( rat > nat ) > set_rat > set_nat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Rat__Orat,type,
image_rat_rat: ( rat > rat ) > set_rat > set_rat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Set__Oset_It__Nat__Onat_J,type,
image_rat_set_nat: ( rat > set_nat ) > set_rat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Set__Oset_It__Rat__Orat_J,type,
image_rat_set_rat: ( rat > set_rat ) > set_rat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Set__Oset_It__Syntax__Orule_J,type,
image_rat_set_rule: ( rat > set_rule ) > set_rat > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Syntax__Orule,type,
image_rat_rule: ( rat > rule ) > set_rat > set_rule ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_4408659257933336347et_rat: ( set_nat > set_rat ) > set_set_nat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Syntax__Orule_J,type,
image_458447791132712456t_rule: ( set_nat > set_rule ) > set_set_nat > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Nat__Onat,type,
image_set_rat_nat: ( set_rat > nat ) > set_set_rat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Rat__Orat,type,
image_set_rat_rat: ( set_rat > rat ) > set_set_rat > set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_3939399684171694371et_rat: ( set_rat > set_rat ) > set_set_rat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Syntax__Orule,type,
image_set_rat_rule: ( set_rat > rule ) > set_set_rat > set_rule ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Syntax__Orule_J_001t__Nat__Onat,type,
image_set_rule_nat: ( set_rule > nat ) > set_set_rule > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Syntax__Orule_J_001t__Set__Oset_It__Syntax__Orule_J,type,
image_2455769455774476541t_rule: ( set_rule > set_rule ) > set_set_rule > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Stream__Ostream_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7912102293542740589et_nat: ( stream_nat > set_nat ) > set_stream_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Stream__Ostream_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_3934614161387701885et_rat: ( stream_rat > set_rat ) > set_stream_rat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Stream__Ostream_It__Syntax__Orule_J_001t__Set__Oset_It__Syntax__Orule_J,type,
image_6459725099818367575t_rule: ( stream_rule > set_rule ) > set_stream_rule > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Nat__Onat,type,
image_rule_nat: ( rule > nat ) > set_rule > set_nat ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Rat__Orat,type,
image_rule_rat: ( rule > rat ) > set_rule > set_rat ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Set__Oset_It__Nat__Onat_J,type,
image_rule_set_nat: ( rule > set_nat ) > set_rule > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Set__Oset_It__Rat__Orat_J,type,
image_rule_set_rat: ( rule > set_rat ) > set_rule > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Set__Oset_It__Syntax__Orule_J,type,
image_rule_set_rule: ( rule > set_rule ) > set_rule > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Syntax__Orule,type,
image_rule_rule: ( rule > rule ) > set_rule > set_rule ).
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__Rat__Orat,type,
set_or4029947393144176647an_rat: rat > rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Rat__Orat_J,type,
set_or32047845639629757et_rat: set_rat > set_rat > set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Syntax__Orule_J,type,
set_or4567957997179852970t_rule: set_rule > set_rule > set_set_rule ).
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__Rat__Orat,type,
set_ord_atLeast_rat: rat > set_rat ).
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__Rat__Orat_J,type,
set_or7446828528931440131et_rat: set_rat > set_set_rat ).
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_It__Rat__Orat_J_J,type,
set_or4639696602114667193et_rat: set_set_rat > set_set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
set_or113368889282349094t_rule: set_set_rule > set_set_set_rule ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Syntax__Orule_J,type,
set_or5121885173736058096t_rule: set_rule > set_set_rule ).
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__Rat__Orat,type,
set_ord_atMost_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Rat__Orat_J,type,
set_or728397472755099399et_rat: set_rat > set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or7210490968680142261et_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_or830504631202464701et_rat: set_set_rat > set_set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
set_or4567209953059919146t_rule: set_set_rule > set_set_set_rule ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Syntax__Orule_J,type,
set_or7438564873865873908t_rule: set_rule > set_set_rule ).
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_001t__Rat__Orat,type,
set_or575021546402375026an_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or458868116921152288et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Rat__Orat_J,type,
set_or6174011595382531368et_rat: set_rat > set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or3831215249870393302et_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_or6674600949247491550et_rat: set_set_rat > set_set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
set_or6920019677468954187t_rule: set_set_rule > set_set_set_rule ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Syntax__Orule_J,type,
set_or3245591702714615573t_rule: set_rule > set_set_rule ).
thf(sy_c_Stream_Osmember_001t__Nat__Onat,type,
smember_nat: nat > stream_nat > $o ).
thf(sy_c_Stream_Osmember_001t__Rat__Orat,type,
smember_rat: rat > stream_rat > $o ).
thf(sy_c_Stream_Osmember_001t__Syntax__Orule,type,
smember_rule: rule > stream_rule > $o ).
thf(sy_c_Stream_Osmerge_001t__Nat__Onat,type,
smerge_nat: stream_stream_nat > stream_nat ).
thf(sy_c_Stream_Osmerge_001t__Rat__Orat,type,
smerge_rat: stream_stream_rat > stream_rat ).
thf(sy_c_Stream_Osmerge_001t__Syntax__Orule,type,
smerge_rule: stream_stream_rule > stream_rule ).
thf(sy_c_Stream_Ostream_OSCons_001t__Nat__Onat,type,
sCons_nat: nat > stream_nat > stream_nat ).
thf(sy_c_Stream_Ostream_OSCons_001t__Rat__Orat,type,
sCons_rat: rat > stream_rat > stream_rat ).
thf(sy_c_Stream_Ostream_OSCons_001t__Syntax__Orule,type,
sCons_rule: rule > stream_rule > stream_rule ).
thf(sy_c_Stream_Ostream_Opred__stream_001t__Nat__Onat,type,
pred_stream_nat: ( nat > $o ) > stream_nat > $o ).
thf(sy_c_Stream_Ostream_Opred__stream_001t__Rat__Orat,type,
pred_stream_rat: ( rat > $o ) > stream_rat > $o ).
thf(sy_c_Stream_Ostream_Opred__stream_001t__Syntax__Orule,type,
pred_stream_rule: ( rule > $o ) > stream_rule > $o ).
thf(sy_c_Stream_Ostream_Osset_001t__Nat__Onat,type,
sset_nat: stream_nat > set_nat ).
thf(sy_c_Stream_Ostream_Osset_001t__Rat__Orat,type,
sset_rat: stream_rat > set_rat ).
thf(sy_c_Stream_Ostream_Osset_001t__Set__Oset_It__Nat__Onat_J,type,
sset_set_nat: stream_set_nat > set_set_nat ).
thf(sy_c_Stream_Ostream_Osset_001t__Set__Oset_It__Rat__Orat_J,type,
sset_set_rat: stream_set_rat > set_set_rat ).
thf(sy_c_Stream_Ostream_Osset_001t__Set__Oset_It__Syntax__Orule_J,type,
sset_set_rule: stream_set_rule > set_set_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__Stream__Ostream_It__Nat__Onat_J,type,
sset_stream_nat: stream_stream_nat > set_stream_nat ).
thf(sy_c_Stream_Ostream_Osset_001t__Stream__Ostream_It__Rat__Orat_J,type,
sset_stream_rat: stream_stream_rat > set_stream_rat ).
thf(sy_c_Stream_Ostream_Osset_001t__Stream__Ostream_It__Syntax__Orule_J,type,
sset_stream_rule: stream_stream_rule > set_stream_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__Syntax__Orule,type,
sset_rule: stream_rule > set_rule ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $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__Rat__Orat_J,type,
member_set_rat: set_rat > set_set_rat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
member_set_set_rat: set_set_rat > set_set_set_rat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
member_set_set_rule: set_set_rule > set_set_set_rule > $o ).
thf(sy_c_member_001t__Set__Oset_It__Syntax__Orule_J,type,
member_set_rule: set_rule > set_set_rule > $o ).
thf(sy_c_member_001t__Syntax__Orule,type,
member_rule: rule > set_rule > $o ).
% Relevant facts (1275)
thf(fact_0_UNIV__I,axiom,
! [X: set_rat] : ( member_set_rat @ X @ top_top_set_set_rat ) ).
% UNIV_I
thf(fact_1_UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% UNIV_I
thf(fact_2_UNIV__I,axiom,
! [X: set_rule] : ( member_set_rule @ X @ top_top_set_set_rule ) ).
% UNIV_I
thf(fact_3_UNIV__I,axiom,
! [X: rule] : ( member_rule @ X @ top_top_set_rule ) ).
% UNIV_I
thf(fact_4_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_5_UNIV__I,axiom,
! [X: rat] : ( member_rat @ X @ top_top_set_rat ) ).
% UNIV_I
thf(fact_6_iso__tuple__UNIV__I,axiom,
! [X: set_rat] : ( member_set_rat @ X @ top_top_set_set_rat ) ).
% iso_tuple_UNIV_I
thf(fact_7_iso__tuple__UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_8_iso__tuple__UNIV__I,axiom,
! [X: set_rule] : ( member_set_rule @ X @ top_top_set_set_rule ) ).
% iso_tuple_UNIV_I
thf(fact_9_iso__tuple__UNIV__I,axiom,
! [X: rule] : ( member_rule @ X @ top_top_set_rule ) ).
% iso_tuple_UNIV_I
thf(fact_10_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_11_iso__tuple__UNIV__I,axiom,
! [X: rat] : ( member_rat @ X @ top_top_set_rat ) ).
% iso_tuple_UNIV_I
thf(fact_12_rules__def,axiom,
( rules
= ( fair_f4564919574533178778m_rule @ rule_of_nat ) ) ).
% rules_def
thf(fact_13_UNIV__eq__I,axiom,
! [A: set_set_rat] :
( ! [X2: set_rat] : ( member_set_rat @ X2 @ A )
=> ( top_top_set_set_rat = A ) ) ).
% UNIV_eq_I
thf(fact_14_UNIV__eq__I,axiom,
! [A: set_set_nat] :
( ! [X2: set_nat] : ( member_set_nat @ X2 @ A )
=> ( top_top_set_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_15_UNIV__eq__I,axiom,
! [A: set_set_rule] :
( ! [X2: set_rule] : ( member_set_rule @ X2 @ A )
=> ( top_top_set_set_rule = A ) ) ).
% UNIV_eq_I
thf(fact_16_UNIV__eq__I,axiom,
! [A: set_rule] :
( ! [X2: rule] : ( member_rule @ X2 @ A )
=> ( top_top_set_rule = A ) ) ).
% UNIV_eq_I
thf(fact_17_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_18_UNIV__eq__I,axiom,
! [A: set_rat] :
( ! [X2: rat] : ( member_rat @ X2 @ A )
=> ( top_top_set_rat = A ) ) ).
% UNIV_eq_I
thf(fact_19_UNIV__witness,axiom,
? [X2: set_rat] : ( member_set_rat @ X2 @ top_top_set_set_rat ) ).
% UNIV_witness
thf(fact_20_UNIV__witness,axiom,
? [X2: set_nat] : ( member_set_nat @ X2 @ top_top_set_set_nat ) ).
% UNIV_witness
thf(fact_21_UNIV__witness,axiom,
? [X2: set_rule] : ( member_set_rule @ X2 @ top_top_set_set_rule ) ).
% UNIV_witness
thf(fact_22_UNIV__witness,axiom,
? [X2: rule] : ( member_rule @ X2 @ top_top_set_rule ) ).
% UNIV_witness
thf(fact_23_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_24_UNIV__witness,axiom,
? [X2: rat] : ( member_rat @ X2 @ top_top_set_rat ) ).
% UNIV_witness
thf(fact_25_Stream_Osmember__def,axiom,
( smember_rat
= ( ^ [X3: rat,S: stream_rat] : ( member_rat @ X3 @ ( sset_rat @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_26_Stream_Osmember__def,axiom,
( smember_nat
= ( ^ [X3: nat,S: stream_nat] : ( member_nat @ X3 @ ( sset_nat @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_27_Stream_Osmember__def,axiom,
( smember_rule
= ( ^ [X3: rule,S: stream_rule] : ( member_rule @ X3 @ ( sset_rule @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_28_Sup__UNIV,axiom,
( ( comple3392050375588816791et_rat @ top_to535976598521457322et_rat )
= top_top_set_set_rat ) ).
% Sup_UNIV
thf(fact_29_Sup__UNIV,axiom,
( ( comple548664676211718543et_nat @ top_to1114752748106383522et_nat )
= top_top_set_set_nat ) ).
% Sup_UNIV
thf(fact_30_Sup__UNIV,axiom,
( ( comple4235951075906658052t_rule @ top_to1078124695766848407t_rule )
= top_top_set_set_rule ) ).
% Sup_UNIV
thf(fact_31_Sup__UNIV,axiom,
( ( comple2146307154184993742t_rule @ top_top_set_set_rule )
= top_top_set_rule ) ).
% Sup_UNIV
thf(fact_32_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_33_Sup__UNIV,axiom,
( ( comple3890839924845867745et_rat @ top_top_set_set_rat )
= top_top_set_rat ) ).
% Sup_UNIV
thf(fact_34_atMost__eq__UNIV__iff,axiom,
! [X: set_set_rat] :
( ( ( set_or830504631202464701et_rat @ X )
= top_to535976598521457322et_rat )
= ( X = top_top_set_set_rat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_35_atMost__eq__UNIV__iff,axiom,
! [X: set_set_nat] :
( ( ( set_or7210490968680142261et_nat @ X )
= top_to1114752748106383522et_nat )
= ( X = top_top_set_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_36_atMost__eq__UNIV__iff,axiom,
! [X: set_set_rule] :
( ( ( set_or4567209953059919146t_rule @ X )
= top_to1078124695766848407t_rule )
= ( X = top_top_set_set_rule ) ) ).
% atMost_eq_UNIV_iff
thf(fact_37_atMost__eq__UNIV__iff,axiom,
! [X: set_rule] :
( ( ( set_or7438564873865873908t_rule @ X )
= top_top_set_set_rule )
= ( X = top_top_set_rule ) ) ).
% atMost_eq_UNIV_iff
thf(fact_38_atMost__eq__UNIV__iff,axiom,
! [X: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X )
= top_top_set_set_nat )
= ( X = top_top_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_39_atMost__eq__UNIV__iff,axiom,
! [X: set_rat] :
( ( ( set_or728397472755099399et_rat @ X )
= top_top_set_set_rat )
= ( X = top_top_set_rat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_40_i_Osset__fenum,axiom,
! [Rules: stream_rule] :
( ( sset_rule @ ( abstra7284221463285775110m_rule @ Rules ) )
= ( sset_rule @ Rules ) ) ).
% i.sset_fenum
thf(fact_41_RuleSystem__Defs_Osset__fenum,axiom,
! [Rules: stream_rule] :
( ( sset_rule @ ( abstra7284221463285775110m_rule @ Rules ) )
= ( sset_rule @ Rules ) ) ).
% RuleSystem_Defs.sset_fenum
thf(fact_42_atMost__UNIV__triv,axiom,
( ( set_or830504631202464701et_rat @ top_top_set_set_rat )
= top_to535976598521457322et_rat ) ).
% atMost_UNIV_triv
thf(fact_43_atMost__UNIV__triv,axiom,
( ( set_or7210490968680142261et_nat @ top_top_set_set_nat )
= top_to1114752748106383522et_nat ) ).
% atMost_UNIV_triv
thf(fact_44_atMost__UNIV__triv,axiom,
( ( set_or4567209953059919146t_rule @ top_top_set_set_rule )
= top_to1078124695766848407t_rule ) ).
% atMost_UNIV_triv
thf(fact_45_atMost__UNIV__triv,axiom,
( ( set_or728397472755099399et_rat @ top_top_set_rat )
= top_top_set_set_rat ) ).
% atMost_UNIV_triv
thf(fact_46_atMost__UNIV__triv,axiom,
( ( set_or4236626031148496127et_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% atMost_UNIV_triv
thf(fact_47_atMost__UNIV__triv,axiom,
( ( set_or7438564873865873908t_rule @ top_top_set_rule )
= top_top_set_set_rule ) ).
% atMost_UNIV_triv
thf(fact_48_atMost__eq__iff,axiom,
! [X: set_rat,Y: set_rat] :
( ( ( set_or728397472755099399et_rat @ X )
= ( set_or728397472755099399et_rat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_49_atMost__eq__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X )
= ( set_or4236626031148496127et_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_50_atMost__eq__iff,axiom,
! [X: set_rule,Y: set_rule] :
( ( ( set_or7438564873865873908t_rule @ X )
= ( set_or7438564873865873908t_rule @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_51_Sup__atMost,axiom,
! [Y: set_rat] :
( ( comple3890839924845867745et_rat @ ( set_or728397472755099399et_rat @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_52_Sup__atMost,axiom,
! [Y: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_53_Sup__atMost,axiom,
! [Y: set_rule] :
( ( comple2146307154184993742t_rule @ ( set_or7438564873865873908t_rule @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_54_top__set__def,axiom,
( top_top_set_rule
= ( collect_rule @ top_top_rule_o ) ) ).
% top_set_def
thf(fact_55_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_56_top__set__def,axiom,
( top_top_set_rat
= ( collect_rat @ top_top_rat_o ) ) ).
% top_set_def
thf(fact_57_top__set__def,axiom,
( top_top_set_set_rat
= ( collect_set_rat @ top_top_set_rat_o ) ) ).
% top_set_def
thf(fact_58_top__set__def,axiom,
( top_top_set_set_nat
= ( collect_set_nat @ top_top_set_nat_o ) ) ).
% top_set_def
thf(fact_59_top__set__def,axiom,
( top_top_set_set_rule
= ( collect_set_rule @ top_top_set_rule_o ) ) ).
% top_set_def
thf(fact_60_Union__UNIV,axiom,
( ( comple3392050375588816791et_rat @ top_to535976598521457322et_rat )
= top_top_set_set_rat ) ).
% Union_UNIV
thf(fact_61_Union__UNIV,axiom,
( ( comple548664676211718543et_nat @ top_to1114752748106383522et_nat )
= top_top_set_set_nat ) ).
% Union_UNIV
thf(fact_62_Union__UNIV,axiom,
( ( comple4235951075906658052t_rule @ top_to1078124695766848407t_rule )
= top_top_set_set_rule ) ).
% Union_UNIV
thf(fact_63_Union__UNIV,axiom,
( ( comple3890839924845867745et_rat @ top_top_set_set_rat )
= top_top_set_rat ) ).
% Union_UNIV
thf(fact_64_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_65_Union__UNIV,axiom,
( ( comple2146307154184993742t_rule @ top_top_set_set_rule )
= top_top_set_rule ) ).
% Union_UNIV
thf(fact_66_not__UNIV__eq__Iic,axiom,
! [H: nat] :
( top_top_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_UNIV_eq_Iic
thf(fact_67_not__UNIV__eq__Iic,axiom,
! [H: rat] :
( top_top_set_rat
!= ( set_ord_atMost_rat @ H ) ) ).
% not_UNIV_eq_Iic
thf(fact_68_cSup__atMost,axiom,
! [X: set_rat] :
( ( comple3890839924845867745et_rat @ ( set_or728397472755099399et_rat @ X ) )
= X ) ).
% cSup_atMost
thf(fact_69_cSup__atMost,axiom,
! [X: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ X ) )
= X ) ).
% cSup_atMost
thf(fact_70_cSup__atMost,axiom,
! [X: set_rule] :
( ( comple2146307154184993742t_rule @ ( set_or7438564873865873908t_rule @ X ) )
= X ) ).
% cSup_atMost
thf(fact_71_top__empty__eq,axiom,
( top_top_rule_o
= ( ^ [X3: rule] : ( member_rule @ X3 @ top_top_set_rule ) ) ) ).
% top_empty_eq
thf(fact_72_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_73_top__empty__eq,axiom,
( top_top_rat_o
= ( ^ [X3: rat] : ( member_rat @ X3 @ top_top_set_rat ) ) ) ).
% top_empty_eq
thf(fact_74_top__empty__eq,axiom,
( top_top_set_rat_o
= ( ^ [X3: set_rat] : ( member_set_rat @ X3 @ top_top_set_set_rat ) ) ) ).
% top_empty_eq
thf(fact_75_top__empty__eq,axiom,
( top_top_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ top_top_set_set_nat ) ) ) ).
% top_empty_eq
thf(fact_76_top__empty__eq,axiom,
( top_top_set_rule_o
= ( ^ [X3: set_rule] : ( member_set_rule @ X3 @ top_top_set_set_rule ) ) ) ).
% top_empty_eq
thf(fact_77_surj__rule__of__nat,axiom,
( ( image_nat_rule @ rule_of_nat @ top_top_set_nat )
= top_top_set_rule ) ).
% surj_rule_of_nat
thf(fact_78_Sup__atLeast,axiom,
! [X: set_set_rat] :
( ( comple3392050375588816791et_rat @ ( set_or4639696602114667193et_rat @ X ) )
= top_top_set_set_rat ) ).
% Sup_atLeast
thf(fact_79_Sup__atLeast,axiom,
! [X: set_set_nat] :
( ( comple548664676211718543et_nat @ ( set_or1796310902737568945et_nat @ X ) )
= top_top_set_set_nat ) ).
% Sup_atLeast
thf(fact_80_Sup__atLeast,axiom,
! [X: set_set_rule] :
( ( comple4235951075906658052t_rule @ ( set_or113368889282349094t_rule @ X ) )
= top_top_set_set_rule ) ).
% Sup_atLeast
thf(fact_81_Sup__atLeast,axiom,
! [X: set_rat] :
( ( comple3890839924845867745et_rat @ ( set_or7446828528931440131et_rat @ X ) )
= top_top_set_rat ) ).
% Sup_atLeast
thf(fact_82_Sup__atLeast,axiom,
! [X: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or1731685050470061051et_nat @ X ) )
= top_top_set_nat ) ).
% Sup_atLeast
thf(fact_83_Sup__atLeast,axiom,
! [X: set_rule] :
( ( comple2146307154184993742t_rule @ ( set_or5121885173736058096t_rule @ X ) )
= top_top_set_rule ) ).
% Sup_atLeast
thf(fact_84_UNIV__stream,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( ( sset_rule @ ( fair_f4564919574533178778m_rule @ F ) )
= top_top_set_rule ) ) ).
% UNIV_stream
thf(fact_85_UNIV__stream,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( sset_nat @ ( fair_fair_stream_nat @ F ) )
= top_top_set_nat ) ) ).
% UNIV_stream
thf(fact_86_UNIV__stream,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ( sset_rat @ ( fair_fair_stream_rat @ F ) )
= top_top_set_rat ) ) ).
% UNIV_stream
thf(fact_87_UNIV__stream,axiom,
! [F: nat > set_rat] :
( ( ( image_nat_set_rat @ F @ top_top_set_nat )
= top_top_set_set_rat )
=> ( ( sset_set_rat @ ( fair_f1980322424541659363et_rat @ F ) )
= top_top_set_set_rat ) ) ).
% UNIV_stream
thf(fact_88_UNIV__stream,axiom,
! [F: nat > set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
=> ( ( sset_set_nat @ ( fair_f5488550982935056091et_nat @ F ) )
= top_top_set_set_nat ) ) ).
% UNIV_stream
thf(fact_89_UNIV__stream,axiom,
! [F: nat > set_rule] :
( ( ( image_nat_set_rule @ F @ top_top_set_nat )
= top_top_set_set_rule )
=> ( ( sset_set_rule @ ( fair_f1814642869706786768t_rule @ F ) )
= top_top_set_set_rule ) ) ).
% UNIV_stream
thf(fact_90_smember__code,axiom,
! [X: rule,Y: rule,S2: stream_rule] :
( ( smember_rule @ X @ ( sCons_rule @ Y @ S2 ) )
= ( ( X != Y )
=> ( smember_rule @ X @ S2 ) ) ) ).
% smember_code
thf(fact_91_not__UNIV__le__Iic,axiom,
! [H2: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_ord_atMost_nat @ H2 ) ) ).
% not_UNIV_le_Iic
thf(fact_92_not__UNIV__le__Iic,axiom,
! [H2: rat] :
~ ( ord_less_eq_set_rat @ top_top_set_rat @ ( set_ord_atMost_rat @ H2 ) ) ).
% not_UNIV_le_Iic
thf(fact_93_stream_Opred__mono__strong,axiom,
! [P: rat > $o,X: stream_rat,Pa: rat > $o] :
( ( pred_stream_rat @ P @ X )
=> ( ! [Z: rat] :
( ( member_rat @ Z @ ( sset_rat @ X ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( pred_stream_rat @ Pa @ X ) ) ) ).
% stream.pred_mono_strong
thf(fact_94_stream_Opred__mono__strong,axiom,
! [P: nat > $o,X: stream_nat,Pa: nat > $o] :
( ( pred_stream_nat @ P @ X )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( sset_nat @ X ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( pred_stream_nat @ Pa @ X ) ) ) ).
% stream.pred_mono_strong
thf(fact_95_stream_Opred__mono__strong,axiom,
! [P: rule > $o,X: stream_rule,Pa: rule > $o] :
( ( pred_stream_rule @ P @ X )
=> ( ! [Z: rule] :
( ( member_rule @ Z @ ( sset_rule @ X ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( pred_stream_rule @ Pa @ X ) ) ) ).
% stream.pred_mono_strong
thf(fact_96_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_97_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_98_image__eqI,axiom,
! [B: rat,F: rat > rat,X: rat,A: set_rat] :
( ( B
= ( F @ X ) )
=> ( ( member_rat @ X @ A )
=> ( member_rat @ B @ ( image_rat_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_99_image__eqI,axiom,
! [B: nat,F: rat > nat,X: rat,A: set_rat] :
( ( B
= ( F @ X ) )
=> ( ( member_rat @ X @ A )
=> ( member_nat @ B @ ( image_rat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_100_image__eqI,axiom,
! [B: rule,F: rat > rule,X: rat,A: set_rat] :
( ( B
= ( F @ X ) )
=> ( ( member_rat @ X @ A )
=> ( member_rule @ B @ ( image_rat_rule @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_101_image__eqI,axiom,
! [B: rat,F: nat > rat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_rat @ B @ ( image_nat_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_102_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_103_image__eqI,axiom,
! [B: rule,F: nat > rule,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_rule @ B @ ( image_nat_rule @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_104_image__eqI,axiom,
! [B: rat,F: rule > rat,X: rule,A: set_rule] :
( ( B
= ( F @ X ) )
=> ( ( member_rule @ X @ A )
=> ( member_rat @ B @ ( image_rule_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_105_image__eqI,axiom,
! [B: nat,F: rule > nat,X: rule,A: set_rule] :
( ( B
= ( F @ X ) )
=> ( ( member_rule @ X @ A )
=> ( member_nat @ B @ ( image_rule_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_106_image__eqI,axiom,
! [B: rule,F: rule > rule,X: rule,A: set_rule] :
( ( B
= ( F @ X ) )
=> ( ( member_rule @ X @ A )
=> ( member_rule @ B @ ( image_rule_rule @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_107_subsetI,axiom,
! [A: set_rat,B2: set_rat] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_rat @ X2 @ B2 ) )
=> ( ord_less_eq_set_rat @ A @ B2 ) ) ).
% subsetI
thf(fact_108_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_109_subsetI,axiom,
! [A: set_rule,B2: set_rule] :
( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( member_rule @ X2 @ B2 ) )
=> ( ord_less_eq_set_rule @ A @ B2 ) ) ).
% subsetI
thf(fact_110_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_rat,P: rat > $o] :
( ( ? [X3: rat] :
( ( member_rat @ X3 @ ( comple3890839924845867745et_rat @ A ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
& ? [Y2: rat] :
( ( member_rat @ Y2 @ X3 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_111_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ A ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ X3 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_112_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_rule,P: rule > $o] :
( ( ? [X3: rule] :
( ( member_rule @ X3 @ ( comple2146307154184993742t_rule @ A ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_rule] :
( ( member_set_rule @ X3 @ A )
& ? [Y2: rule] :
( ( member_rule @ Y2 @ X3 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_113_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_rat,P: rat > $o] :
( ( ! [X3: rat] :
( ( member_rat @ X3 @ ( comple3890839924845867745et_rat @ A ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
=> ! [Y2: rat] :
( ( member_rat @ Y2 @ X3 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_114_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ A ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ X3 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_115_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_rule,P: rule > $o] :
( ( ! [X3: rule] :
( ( member_rule @ X3 @ ( comple2146307154184993742t_rule @ A ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_rule] :
( ( member_set_rule @ X3 @ A )
=> ! [Y2: rule] :
( ( member_rule @ Y2 @ X3 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_116_UnionI,axiom,
! [X4: set_rat,C: set_set_rat,A: rat] :
( ( member_set_rat @ X4 @ C )
=> ( ( member_rat @ A @ X4 )
=> ( member_rat @ A @ ( comple3890839924845867745et_rat @ C ) ) ) ) ).
% UnionI
thf(fact_117_UnionI,axiom,
! [X4: set_nat,C: set_set_nat,A: nat] :
( ( member_set_nat @ X4 @ C )
=> ( ( member_nat @ A @ X4 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_118_UnionI,axiom,
! [X4: set_rule,C: set_set_rule,A: rule] :
( ( member_set_rule @ X4 @ C )
=> ( ( member_rule @ A @ X4 )
=> ( member_rule @ A @ ( comple2146307154184993742t_rule @ C ) ) ) ) ).
% UnionI
thf(fact_119_Union__iff,axiom,
! [A: rat,C: set_set_rat] :
( ( member_rat @ A @ ( comple3890839924845867745et_rat @ C ) )
= ( ? [X3: set_rat] :
( ( member_set_rat @ X3 @ C )
& ( member_rat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_120_Union__iff,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ C )
& ( member_nat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_121_Union__iff,axiom,
! [A: rule,C: set_set_rule] :
( ( member_rule @ A @ ( comple2146307154184993742t_rule @ C ) )
= ( ? [X3: set_rule] :
( ( member_set_rule @ X3 @ C )
& ( member_rule @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_122_atMost__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
= ( ord_less_eq_rat @ I @ K ) ) ).
% atMost_iff
thf(fact_123_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_124_atMost__iff,axiom,
! [I: set_rat,K: set_rat] :
( ( member_set_rat @ I @ ( set_or728397472755099399et_rat @ K ) )
= ( ord_less_eq_set_rat @ I @ K ) ) ).
% atMost_iff
thf(fact_125_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_126_atMost__iff,axiom,
! [I: set_rule,K: set_rule] :
( ( member_set_rule @ I @ ( set_or7438564873865873908t_rule @ K ) )
= ( ord_less_eq_set_rule @ I @ K ) ) ).
% atMost_iff
thf(fact_127_mem__Collect__eq,axiom,
! [A2: rat,P: rat > $o] :
( ( member_rat @ A2 @ ( collect_rat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_129_mem__Collect__eq,axiom,
! [A2: rule,P: rule > $o] :
( ( member_rule @ A2 @ ( collect_rule @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A: set_rat] :
( ( collect_rat
@ ^ [X3: rat] : ( member_rat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_132_Collect__mem__eq,axiom,
! [A: set_rule] :
( ( collect_rule
@ ^ [X3: rule] : ( member_rule @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_133_atLeast__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_ord_atLeast_rat @ K ) )
= ( ord_less_eq_rat @ K @ I ) ) ).
% atLeast_iff
thf(fact_134_atLeast__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atLeast_nat @ K ) )
= ( ord_less_eq_nat @ K @ I ) ) ).
% atLeast_iff
thf(fact_135_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_136_atMost__subset__iff,axiom,
! [X: set_rat,Y: set_rat] :
( ( ord_le513522071413781156et_rat @ ( set_or728397472755099399et_rat @ X ) @ ( set_or728397472755099399et_rat @ Y ) )
= ( ord_less_eq_set_rat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_137_atMost__subset__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_138_atMost__subset__iff,axiom,
! [X: set_rule,Y: set_rule] :
( ( ord_le7968974978423766289t_rule @ ( set_or7438564873865873908t_rule @ X ) @ ( set_or7438564873865873908t_rule @ Y ) )
= ( ord_less_eq_set_rule @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_139_atLeast__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ X ) @ ( set_ord_atLeast_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_140_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_141_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_142_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_143_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > rule,D: nat > rule,Inf: set_rule > rule] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_rule @ C @ A ) )
= ( Inf @ ( image_nat_rule @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_144_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > rat,D: nat > rat,Inf: set_rat > rat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_rat @ C @ A ) )
= ( Inf @ ( image_nat_rat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_145_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > rule,D: nat > rule,Sup: set_rule > rule] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_rule @ C @ A ) )
= ( Sup @ ( image_nat_rule @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_146_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > rat,D: nat > rat,Sup: set_rat > rat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_rat @ C @ A ) )
= ( Sup @ ( image_nat_rat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_147_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_148_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_149_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_150_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_151_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_152_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_153_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_154_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_155_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_156_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_157_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_158_order_Otrans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_159_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_160_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_161_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_162_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_163_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_164_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: 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 @ B ) ) ) ).
% linorder_wlog
thf(fact_165_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_166_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_167_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_168_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_169_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_170_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_171_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_172_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_173_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_174_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: 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 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_175_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_176_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_177_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_178_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_179_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_180_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_181_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_182_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_183_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_184_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_185_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_186_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_187_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_188_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_189_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_190_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_191_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_192_imageI,axiom,
! [X: rat,A: set_rat,F: rat > rat] :
( ( member_rat @ X @ A )
=> ( member_rat @ ( F @ X ) @ ( image_rat_rat @ F @ A ) ) ) ).
% imageI
thf(fact_193_imageI,axiom,
! [X: rat,A: set_rat,F: rat > nat] :
( ( member_rat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_rat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_194_imageI,axiom,
! [X: rat,A: set_rat,F: rat > rule] :
( ( member_rat @ X @ A )
=> ( member_rule @ ( F @ X ) @ ( image_rat_rule @ F @ A ) ) ) ).
% imageI
thf(fact_195_imageI,axiom,
! [X: nat,A: set_nat,F: nat > rat] :
( ( member_nat @ X @ A )
=> ( member_rat @ ( F @ X ) @ ( image_nat_rat @ F @ A ) ) ) ).
% imageI
thf(fact_196_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_197_imageI,axiom,
! [X: nat,A: set_nat,F: nat > rule] :
( ( member_nat @ X @ A )
=> ( member_rule @ ( F @ X ) @ ( image_nat_rule @ F @ A ) ) ) ).
% imageI
thf(fact_198_imageI,axiom,
! [X: rule,A: set_rule,F: rule > rat] :
( ( member_rule @ X @ A )
=> ( member_rat @ ( F @ X ) @ ( image_rule_rat @ F @ A ) ) ) ).
% imageI
thf(fact_199_imageI,axiom,
! [X: rule,A: set_rule,F: rule > nat] :
( ( member_rule @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_rule_nat @ F @ A ) ) ) ).
% imageI
thf(fact_200_imageI,axiom,
! [X: rule,A: set_rule,F: rule > rule] :
( ( member_rule @ X @ A )
=> ( member_rule @ ( F @ X ) @ ( image_rule_rule @ F @ A ) ) ) ).
% imageI
thf(fact_201_in__mono,axiom,
! [A: set_rat,B2: set_rat,X: rat] :
( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ( member_rat @ X @ A )
=> ( member_rat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_202_in__mono,axiom,
! [A: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_203_in__mono,axiom,
! [A: set_rule,B2: set_rule,X: rule] :
( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ( member_rule @ X @ A )
=> ( member_rule @ X @ B2 ) ) ) ).
% in_mono
thf(fact_204_subsetD,axiom,
! [A: set_rat,B2: set_rat,C2: rat] :
( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ( member_rat @ C2 @ A )
=> ( member_rat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_205_subsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_206_subsetD,axiom,
! [A: set_rule,B2: set_rule,C2: rule] :
( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ( member_rule @ C2 @ A )
=> ( member_rule @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_207_image__iff,axiom,
! [Z2: rat,F: nat > rat,A: set_nat] :
( ( member_rat @ Z2 @ ( image_nat_rat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_208_image__iff,axiom,
! [Z2: rule,F: nat > rule,A: set_nat] :
( ( member_rule @ Z2 @ ( image_nat_rule @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_209_subset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A5: set_rat,B5: set_rat] :
! [X3: rat] :
( ( member_rat @ X3 @ A5 )
=> ( member_rat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_210_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_211_subset__eq,axiom,
( ord_less_eq_set_rule
= ( ^ [A5: set_rule,B5: set_rule] :
! [X3: rule] :
( ( member_rule @ X3 @ A5 )
=> ( member_rule @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_212_bex__imageD,axiom,
! [F: nat > rule,A: set_nat,P: rule > $o] :
( ? [X6: rule] :
( ( member_rule @ X6 @ ( image_nat_rule @ F @ A ) )
& ( P @ X6 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_213_bex__imageD,axiom,
! [F: nat > rat,A: set_nat,P: rat > $o] :
( ? [X6: rat] :
( ( member_rat @ X6 @ ( image_nat_rat @ F @ A ) )
& ( P @ X6 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_214_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > rule,G: nat > rule] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_rule @ F @ M2 )
= ( image_nat_rule @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_215_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > rat,G: nat > rat] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_rat @ F @ M2 )
= ( image_nat_rat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_216_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > rule] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ ( image_nat_rule @ F @ B2 ) ) ) ).
% image_mono
thf(fact_217_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > rat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ ( image_nat_rat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_218_subset__iff,axiom,
( ord_less_eq_set_rat
= ( ^ [A5: set_rat,B5: set_rat] :
! [T: rat] :
( ( member_rat @ T @ A5 )
=> ( member_rat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_219_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_220_subset__iff,axiom,
( ord_less_eq_set_rule
= ( ^ [A5: set_rule,B5: set_rule] :
! [T: rule] :
( ( member_rule @ T @ A5 )
=> ( member_rule @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_221_ball__imageD,axiom,
! [F: nat > rule,A: set_nat,P: rule > $o] :
( ! [X2: rule] :
( ( member_rule @ X2 @ ( image_nat_rule @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_222_ball__imageD,axiom,
! [F: nat > rat,A: set_nat,P: rat > $o] :
( ! [X2: rat] :
( ( member_rat @ X2 @ ( image_nat_rat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_223_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_224_image__subsetI,axiom,
! [A: set_rat,F: rat > rat,B2: set_rat] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rat @ ( image_rat_rat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_225_image__subsetI,axiom,
! [A: set_rat,F: rat > nat,B2: set_nat] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_rat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_226_image__subsetI,axiom,
! [A: set_rat,F: rat > rule,B2: set_rule] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rule @ ( image_rat_rule @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_227_image__subsetI,axiom,
! [A: set_nat,F: nat > rat,B2: set_rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_228_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_229_image__subsetI,axiom,
! [A: set_nat,F: nat > rule,B2: set_rule] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_230_image__subsetI,axiom,
! [A: set_rule,F: rule > rat,B2: set_rat] :
( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rat @ ( image_rule_rat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_231_image__subsetI,axiom,
! [A: set_rule,F: rule > nat,B2: set_nat] :
( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_rule_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_232_image__subsetI,axiom,
! [A: set_rule,F: rule > rule,B2: set_rule] :
( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rule @ ( image_rule_rule @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_233_rev__image__eqI,axiom,
! [X: rat,A: set_rat,B: rat,F: rat > rat] :
( ( member_rat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_rat_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_234_rev__image__eqI,axiom,
! [X: rat,A: set_rat,B: nat,F: rat > nat] :
( ( member_rat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_rat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_235_rev__image__eqI,axiom,
! [X: rat,A: set_rat,B: rule,F: rat > rule] :
( ( member_rat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_rat_rule @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_236_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: rat,F: nat > rat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_nat_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_237_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_238_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: rule,F: nat > rule] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_nat_rule @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_239_rev__image__eqI,axiom,
! [X: rule,A: set_rule,B: rat,F: rule > rat] :
( ( member_rule @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_rule_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_240_rev__image__eqI,axiom,
! [X: rule,A: set_rule,B: nat,F: rule > nat] :
( ( member_rule @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_rule_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_241_rev__image__eqI,axiom,
! [X: rule,A: set_rule,B: rule,F: rule > rule] :
( ( member_rule @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_rule_rule @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_242_subset__imageE,axiom,
! [B2: set_rule,F: nat > rule,A: set_nat] :
( ( ord_less_eq_set_rule @ B2 @ ( image_nat_rule @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B2
!= ( image_nat_rule @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_243_subset__imageE,axiom,
! [B2: set_rat,F: nat > rat,A: set_nat] :
( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B2
!= ( image_nat_rat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_244_image__subset__iff,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_rat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_245_image__subset__iff,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_rule @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_246_subset__image__iff,axiom,
! [B2: set_rule,F: nat > rule,A: set_nat] :
( ( ord_less_eq_set_rule @ B2 @ ( image_nat_rule @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_rule @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_247_subset__image__iff,axiom,
! [B2: set_rat,F: nat > rat,A: set_nat] :
( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_rat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_248_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_249_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_250_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_251_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_252_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_253_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_254_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_255_UnionE,axiom,
! [A: rat,C: set_set_rat] :
( ( member_rat @ A @ ( comple3890839924845867745et_rat @ C ) )
=> ~ ! [X7: set_rat] :
( ( member_rat @ A @ X7 )
=> ~ ( member_set_rat @ X7 @ C ) ) ) ).
% UnionE
thf(fact_256_UnionE,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
=> ~ ! [X7: set_nat] :
( ( member_nat @ A @ X7 )
=> ~ ( member_set_nat @ X7 @ C ) ) ) ).
% UnionE
thf(fact_257_UnionE,axiom,
! [A: rule,C: set_set_rule] :
( ( member_rule @ A @ ( comple2146307154184993742t_rule @ C ) )
=> ~ ! [X7: set_rule] :
( ( member_rule @ A @ X7 )
=> ~ ( member_set_rule @ X7 @ C ) ) ) ).
% UnionE
thf(fact_258_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_259_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_260_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_261_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_262_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_263_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_264_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_265_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_266_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_267_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_268_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_269_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_270_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_271_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_272_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_273_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_274_Union__mono,axiom,
! [A: set_set_rat,B2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B2 )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ ( comple3890839924845867745et_rat @ B2 ) ) ) ).
% Union_mono
thf(fact_275_Union__mono,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_276_Union__mono,axiom,
! [A: set_set_rule,B2: set_set_rule] :
( ( ord_le7968974978423766289t_rule @ A @ B2 )
=> ( ord_less_eq_set_rule @ ( comple2146307154184993742t_rule @ A ) @ ( comple2146307154184993742t_rule @ B2 ) ) ) ).
% Union_mono
thf(fact_277_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_278_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_279_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_280_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_281_Union__least,axiom,
! [A: set_set_rat,C: set_rat] :
( ! [X7: set_rat] :
( ( member_set_rat @ X7 @ A )
=> ( ord_less_eq_set_rat @ X7 @ C ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ C ) ) ).
% Union_least
thf(fact_282_Union__least,axiom,
! [A: set_set_nat,C: set_nat] :
( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ A )
=> ( ord_less_eq_set_nat @ X7 @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ C ) ) ).
% Union_least
thf(fact_283_Union__least,axiom,
! [A: set_set_rule,C: set_rule] :
( ! [X7: set_rule] :
( ( member_set_rule @ X7 @ A )
=> ( ord_less_eq_set_rule @ X7 @ C ) )
=> ( ord_less_eq_set_rule @ ( comple2146307154184993742t_rule @ A ) @ C ) ) ).
% Union_least
thf(fact_284_Union__upper,axiom,
! [B2: set_rat,A: set_set_rat] :
( ( member_set_rat @ B2 @ A )
=> ( ord_less_eq_set_rat @ B2 @ ( comple3890839924845867745et_rat @ A ) ) ) ).
% Union_upper
thf(fact_285_Union__upper,axiom,
! [B2: set_nat,A: set_set_nat] :
( ( member_set_nat @ B2 @ A )
=> ( ord_less_eq_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Union_upper
thf(fact_286_Union__upper,axiom,
! [B2: set_rule,A: set_set_rule] :
( ( member_set_rule @ B2 @ A )
=> ( ord_less_eq_set_rule @ B2 @ ( comple2146307154184993742t_rule @ A ) ) ) ).
% Union_upper
thf(fact_287_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_288_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_289_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_290_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_291_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_292_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_293_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_294_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_295_Union__subsetI,axiom,
! [A: set_set_rat,B2: set_set_rat] :
( ! [X2: set_rat] :
( ( member_set_rat @ X2 @ A )
=> ? [Y4: set_rat] :
( ( member_set_rat @ Y4 @ B2 )
& ( ord_less_eq_set_rat @ X2 @ Y4 ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ ( comple3890839924845867745et_rat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_296_Union__subsetI,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ? [Y4: set_nat] :
( ( member_set_nat @ Y4 @ B2 )
& ( ord_less_eq_set_nat @ X2 @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_297_Union__subsetI,axiom,
! [A: set_set_rule,B2: set_set_rule] :
( ! [X2: set_rule] :
( ( member_set_rule @ X2 @ A )
=> ? [Y4: set_rule] :
( ( member_set_rule @ Y4 @ B2 )
& ( ord_less_eq_set_rule @ X2 @ Y4 ) ) )
=> ( ord_less_eq_set_rule @ ( comple2146307154184993742t_rule @ A ) @ ( comple2146307154184993742t_rule @ B2 ) ) ) ).
% Union_subsetI
thf(fact_298_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_299_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_300_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_301_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_302_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_303_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A2 @ X6 )
& ( ord_less_nat @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D2: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A2 @ X2 )
& ( ord_less_nat @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D2 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_304_range__subsetD,axiom,
! [F: rule > rat,B2: set_rat,I: rule] :
( ( ord_less_eq_set_rat @ ( image_rule_rat @ F @ top_top_set_rule ) @ B2 )
=> ( member_rat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_305_range__subsetD,axiom,
! [F: rule > nat,B2: set_nat,I: rule] :
( ( ord_less_eq_set_nat @ ( image_rule_nat @ F @ top_top_set_rule ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_306_range__subsetD,axiom,
! [F: rule > rule,B2: set_rule,I: rule] :
( ( ord_less_eq_set_rule @ ( image_rule_rule @ F @ top_top_set_rule ) @ B2 )
=> ( member_rule @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_307_range__subsetD,axiom,
! [F: nat > rat,B2: set_rat,I: nat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ top_top_set_nat ) @ B2 )
=> ( member_rat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_308_range__subsetD,axiom,
! [F: nat > nat,B2: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_309_range__subsetD,axiom,
! [F: nat > rule,B2: set_rule,I: nat] :
( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ top_top_set_nat ) @ B2 )
=> ( member_rule @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_310_range__subsetD,axiom,
! [F: rat > rat,B2: set_rat,I: rat] :
( ( ord_less_eq_set_rat @ ( image_rat_rat @ F @ top_top_set_rat ) @ B2 )
=> ( member_rat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_311_range__subsetD,axiom,
! [F: rat > nat,B2: set_nat,I: rat] :
( ( ord_less_eq_set_nat @ ( image_rat_nat @ F @ top_top_set_rat ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_312_range__subsetD,axiom,
! [F: rat > rule,B2: set_rule,I: rat] :
( ( ord_less_eq_set_rule @ ( image_rat_rule @ F @ top_top_set_rat ) @ B2 )
=> ( member_rule @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_313_range__subsetD,axiom,
! [F: set_rat > rat,B2: set_rat,I: set_rat] :
( ( ord_less_eq_set_rat @ ( image_set_rat_rat @ F @ top_top_set_set_rat ) @ B2 )
=> ( member_rat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_314_SUP__eq,axiom,
! [A: set_rat,B2: set_rat,F: rat > set_rat,G: rat > set_rat] :
( ! [I2: rat] :
( ( member_rat @ I2 @ A )
=> ? [X6: rat] :
( ( member_rat @ X6 @ B2 )
& ( ord_less_eq_set_rat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: rat] :
( ( member_rat @ J @ B2 )
=> ? [X6: rat] :
( ( member_rat @ X6 @ A )
& ( ord_less_eq_set_rat @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_315_SUP__eq,axiom,
! [A: set_rat,B2: set_nat,F: rat > set_rat,G: nat > set_rat] :
( ! [I2: rat] :
( ( member_rat @ I2 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_set_rat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B2 )
=> ? [X6: rat] :
( ( member_rat @ X6 @ A )
& ( ord_less_eq_set_rat @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) )
= ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_316_SUP__eq,axiom,
! [A: set_rat,B2: set_rule,F: rat > set_rat,G: rule > set_rat] :
( ! [I2: rat] :
( ( member_rat @ I2 @ A )
=> ? [X6: rule] :
( ( member_rule @ X6 @ B2 )
& ( ord_less_eq_set_rat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: rule] :
( ( member_rule @ J @ B2 )
=> ? [X6: rat] :
( ( member_rat @ X6 @ A )
& ( ord_less_eq_set_rat @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) )
= ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_317_SUP__eq,axiom,
! [A: set_nat,B2: set_rat,F: nat > set_rat,G: rat > set_rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ? [X6: rat] :
( ( member_rat @ X6 @ B2 )
& ( ord_less_eq_set_rat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: rat] :
( ( member_rat @ J @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_set_rat @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ F @ A ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_318_SUP__eq,axiom,
! [A: set_nat,B2: set_nat,F: nat > set_rat,G: nat > set_rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_set_rat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_set_rat @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ F @ A ) )
= ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_319_SUP__eq,axiom,
! [A: set_nat,B2: set_rule,F: nat > set_rat,G: rule > set_rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ? [X6: rule] :
( ( member_rule @ X6 @ B2 )
& ( ord_less_eq_set_rat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: rule] :
( ( member_rule @ J @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_set_rat @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ F @ A ) )
= ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_320_SUP__eq,axiom,
! [A: set_rule,B2: set_rat,F: rule > set_rat,G: rat > set_rat] :
( ! [I2: rule] :
( ( member_rule @ I2 @ A )
=> ? [X6: rat] :
( ( member_rat @ X6 @ B2 )
& ( ord_less_eq_set_rat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: rat] :
( ( member_rat @ J @ B2 )
=> ? [X6: rule] :
( ( member_rule @ X6 @ A )
& ( ord_less_eq_set_rat @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ F @ A ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_321_SUP__eq,axiom,
! [A: set_rule,B2: set_nat,F: rule > set_rat,G: nat > set_rat] :
( ! [I2: rule] :
( ( member_rule @ I2 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_set_rat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B2 )
=> ? [X6: rule] :
( ( member_rule @ X6 @ A )
& ( ord_less_eq_set_rat @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ F @ A ) )
= ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_322_SUP__eq,axiom,
! [A: set_rule,B2: set_rule,F: rule > set_rat,G: rule > set_rat] :
( ! [I2: rule] :
( ( member_rule @ I2 @ A )
=> ? [X6: rule] :
( ( member_rule @ X6 @ B2 )
& ( ord_less_eq_set_rat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: rule] :
( ( member_rule @ J @ B2 )
=> ? [X6: rule] :
( ( member_rule @ X6 @ A )
& ( ord_less_eq_set_rat @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ F @ A ) )
= ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_323_SUP__eq,axiom,
! [A: set_rat,B2: set_rat,F: rat > set_nat,G: rat > set_nat] :
( ! [I2: rat] :
( ( member_rat @ I2 @ A )
=> ? [X6: rat] :
( ( member_rat @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: rat] :
( ( member_rat @ J @ B2 )
=> ? [X6: rat] :
( ( member_rat @ X6 @ A )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_324_not__UNIV__le__Ici,axiom,
! [L: rat] :
~ ( ord_less_eq_set_rat @ top_top_set_rat @ ( set_ord_atLeast_rat @ L ) ) ).
% not_UNIV_le_Ici
thf(fact_325_Sup__subset__mono,axiom,
! [A: set_set_rat,B2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B2 )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ ( comple3890839924845867745et_rat @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_326_Sup__subset__mono,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_327_Sup__subset__mono,axiom,
! [A: set_set_rule,B2: set_set_rule] :
( ( ord_le7968974978423766289t_rule @ A @ B2 )
=> ( ord_less_eq_set_rule @ ( comple2146307154184993742t_rule @ A ) @ ( comple2146307154184993742t_rule @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_328_cSup__eq__maximum,axiom,
! [Z2: nat,X4: set_nat] :
( ( member_nat @ Z2 @ X4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( complete_Sup_Sup_nat @ X4 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_329_cSup__eq__maximum,axiom,
! [Z2: set_rat,X4: set_set_rat] :
( ( member_set_rat @ Z2 @ X4 )
=> ( ! [X2: set_rat] :
( ( member_set_rat @ X2 @ X4 )
=> ( ord_less_eq_set_rat @ X2 @ Z2 ) )
=> ( ( comple3890839924845867745et_rat @ X4 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_330_cSup__eq__maximum,axiom,
! [Z2: set_nat,X4: set_set_nat] :
( ( member_set_nat @ Z2 @ X4 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X4 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) )
=> ( ( comple7399068483239264473et_nat @ X4 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_331_cSup__eq__maximum,axiom,
! [Z2: set_rule,X4: set_set_rule] :
( ( member_set_rule @ Z2 @ X4 )
=> ( ! [X2: set_rule] :
( ( member_set_rule @ X2 @ X4 )
=> ( ord_less_eq_set_rule @ X2 @ Z2 ) )
=> ( ( comple2146307154184993742t_rule @ X4 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_332_rangeI,axiom,
! [F: rule > rat,X: rule] : ( member_rat @ ( F @ X ) @ ( image_rule_rat @ F @ top_top_set_rule ) ) ).
% rangeI
thf(fact_333_rangeI,axiom,
! [F: rule > nat,X: rule] : ( member_nat @ ( F @ X ) @ ( image_rule_nat @ F @ top_top_set_rule ) ) ).
% rangeI
thf(fact_334_rangeI,axiom,
! [F: rule > rule,X: rule] : ( member_rule @ ( F @ X ) @ ( image_rule_rule @ F @ top_top_set_rule ) ) ).
% rangeI
thf(fact_335_rangeI,axiom,
! [F: nat > rat,X: nat] : ( member_rat @ ( F @ X ) @ ( image_nat_rat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_336_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_337_rangeI,axiom,
! [F: nat > rule,X: nat] : ( member_rule @ ( F @ X ) @ ( image_nat_rule @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_338_rangeI,axiom,
! [F: rat > rat,X: rat] : ( member_rat @ ( F @ X ) @ ( image_rat_rat @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_339_rangeI,axiom,
! [F: rat > nat,X: rat] : ( member_nat @ ( F @ X ) @ ( image_rat_nat @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_340_rangeI,axiom,
! [F: rat > rule,X: rat] : ( member_rule @ ( F @ X ) @ ( image_rat_rule @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_341_rangeI,axiom,
! [F: set_rat > rat,X: set_rat] : ( member_rat @ ( F @ X ) @ ( image_set_rat_rat @ F @ top_top_set_set_rat ) ) ).
% rangeI
thf(fact_342_range__eqI,axiom,
! [B: rat,F: rule > rat,X: rule] :
( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_rule_rat @ F @ top_top_set_rule ) ) ) ).
% range_eqI
thf(fact_343_range__eqI,axiom,
! [B: nat,F: rule > nat,X: rule] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_rule_nat @ F @ top_top_set_rule ) ) ) ).
% range_eqI
thf(fact_344_range__eqI,axiom,
! [B: rule,F: rule > rule,X: rule] :
( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_rule_rule @ F @ top_top_set_rule ) ) ) ).
% range_eqI
thf(fact_345_range__eqI,axiom,
! [B: rat,F: nat > rat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_nat_rat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_346_range__eqI,axiom,
! [B: nat,F: nat > nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_347_range__eqI,axiom,
! [B: rule,F: nat > rule,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_nat_rule @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_348_range__eqI,axiom,
! [B: rat,F: rat > rat,X: rat] :
( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_rat_rat @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_349_range__eqI,axiom,
! [B: nat,F: rat > nat,X: rat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_rat_nat @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_350_range__eqI,axiom,
! [B: rule,F: rat > rule,X: rat] :
( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_rat_rule @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_351_range__eqI,axiom,
! [B: rat,F: set_rat > rat,X: set_rat] :
( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_set_rat_rat @ F @ top_top_set_set_rat ) ) ) ).
% range_eqI
thf(fact_352_SUP__cong,axiom,
! [A: set_rat,B2: set_rat,C: rat > set_rat,D: rat > set_rat] :
( ( A = B2 )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C @ A ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_353_SUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > set_rat,D: nat > set_rat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ C @ A ) )
= ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_354_SUP__cong,axiom,
! [A: set_rule,B2: set_rule,C: rule > set_rat,D: rule > set_rat] :
( ( A = B2 )
=> ( ! [X2: rule] :
( ( member_rule @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ C @ A ) )
= ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_355_SUP__cong,axiom,
! [A: set_rat,B2: set_rat,C: rat > set_nat,D: rat > set_nat] :
( ( A = B2 )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ C @ A ) )
= ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_356_SUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > set_nat,D: nat > set_nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_357_SUP__cong,axiom,
! [A: set_rule,B2: set_rule,C: rule > set_nat,D: rule > set_nat] :
( ( A = B2 )
=> ( ! [X2: rule] :
( ( member_rule @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ C @ A ) )
= ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_358_SUP__cong,axiom,
! [A: set_rat,B2: set_rat,C: rat > set_rule,D: rat > set_rule] :
( ( A = B2 )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple2146307154184993742t_rule @ ( image_rat_set_rule @ C @ A ) )
= ( comple2146307154184993742t_rule @ ( image_rat_set_rule @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_359_SUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > set_rule,D: nat > set_rule] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple2146307154184993742t_rule @ ( image_nat_set_rule @ C @ A ) )
= ( comple2146307154184993742t_rule @ ( image_nat_set_rule @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_360_SUP__cong,axiom,
! [A: set_rule,B2: set_rule,C: rule > set_rule,D: rule > set_rule] :
( ( A = B2 )
=> ( ! [X2: rule] :
( ( member_rule @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ C @ A ) )
= ( comple2146307154184993742t_rule @ ( image_rule_set_rule @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_361_top_Oextremum__strict,axiom,
! [A2: set_rule] :
~ ( ord_less_set_rule @ top_top_set_rule @ A2 ) ).
% top.extremum_strict
thf(fact_362_top_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_363_top_Oextremum__strict,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ top_top_set_rat @ A2 ) ).
% top.extremum_strict
thf(fact_364_top_Oextremum__strict,axiom,
! [A2: set_set_rat] :
~ ( ord_less_set_set_rat @ top_top_set_set_rat @ A2 ) ).
% top.extremum_strict
thf(fact_365_top_Oextremum__strict,axiom,
! [A2: set_set_nat] :
~ ( ord_less_set_set_nat @ top_top_set_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_366_top_Oextremum__strict,axiom,
! [A2: set_set_rule] :
~ ( ord_le3472952955546049285t_rule @ top_top_set_set_rule @ A2 ) ).
% top.extremum_strict
thf(fact_367_top_Onot__eq__extremum,axiom,
! [A2: set_rule] :
( ( A2 != top_top_set_rule )
= ( ord_less_set_rule @ A2 @ top_top_set_rule ) ) ).
% top.not_eq_extremum
thf(fact_368_top_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_369_top_Onot__eq__extremum,axiom,
! [A2: set_rat] :
( ( A2 != top_top_set_rat )
= ( ord_less_set_rat @ A2 @ top_top_set_rat ) ) ).
% top.not_eq_extremum
thf(fact_370_top_Onot__eq__extremum,axiom,
! [A2: set_set_rat] :
( ( A2 != top_top_set_set_rat )
= ( ord_less_set_set_rat @ A2 @ top_top_set_set_rat ) ) ).
% top.not_eq_extremum
thf(fact_371_top_Onot__eq__extremum,axiom,
! [A2: set_set_nat] :
( ( A2 != top_top_set_set_nat )
= ( ord_less_set_set_nat @ A2 @ top_top_set_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_372_top_Onot__eq__extremum,axiom,
! [A2: set_set_rule] :
( ( A2 != top_top_set_set_rule )
= ( ord_le3472952955546049285t_rule @ A2 @ top_top_set_set_rule ) ) ).
% top.not_eq_extremum
thf(fact_373_top__greatest,axiom,
! [A2: set_rule] : ( ord_less_eq_set_rule @ A2 @ top_top_set_rule ) ).
% top_greatest
thf(fact_374_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_375_top__greatest,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ top_top_set_rat ) ).
% top_greatest
thf(fact_376_top__greatest,axiom,
! [A2: set_set_rat] : ( ord_le513522071413781156et_rat @ A2 @ top_top_set_set_rat ) ).
% top_greatest
thf(fact_377_top__greatest,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ top_top_set_set_nat ) ).
% top_greatest
thf(fact_378_top__greatest,axiom,
! [A2: set_set_rule] : ( ord_le7968974978423766289t_rule @ A2 @ top_top_set_set_rule ) ).
% top_greatest
thf(fact_379_top_Oextremum__unique,axiom,
! [A2: set_rule] :
( ( ord_less_eq_set_rule @ top_top_set_rule @ A2 )
= ( A2 = top_top_set_rule ) ) ).
% top.extremum_unique
thf(fact_380_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_381_top_Oextremum__unique,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A2 )
= ( A2 = top_top_set_rat ) ) ).
% top.extremum_unique
thf(fact_382_top_Oextremum__unique,axiom,
! [A2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ top_top_set_set_rat @ A2 )
= ( A2 = top_top_set_set_rat ) ) ).
% top.extremum_unique
thf(fact_383_top_Oextremum__unique,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ top_top_set_set_nat @ A2 )
= ( A2 = top_top_set_set_nat ) ) ).
% top.extremum_unique
thf(fact_384_top_Oextremum__unique,axiom,
! [A2: set_set_rule] :
( ( ord_le7968974978423766289t_rule @ top_top_set_set_rule @ A2 )
= ( A2 = top_top_set_set_rule ) ) ).
% top.extremum_unique
thf(fact_385_top_Oextremum__uniqueI,axiom,
! [A2: set_rule] :
( ( ord_less_eq_set_rule @ top_top_set_rule @ A2 )
=> ( A2 = top_top_set_rule ) ) ).
% top.extremum_uniqueI
thf(fact_386_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_387_top_Oextremum__uniqueI,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A2 )
=> ( A2 = top_top_set_rat ) ) ).
% top.extremum_uniqueI
thf(fact_388_top_Oextremum__uniqueI,axiom,
! [A2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ top_top_set_set_rat @ A2 )
=> ( A2 = top_top_set_set_rat ) ) ).
% top.extremum_uniqueI
thf(fact_389_top_Oextremum__uniqueI,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ top_top_set_set_nat @ A2 )
=> ( A2 = top_top_set_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_390_top_Oextremum__uniqueI,axiom,
! [A2: set_set_rule] :
( ( ord_le7968974978423766289t_rule @ top_top_set_set_rule @ A2 )
=> ( A2 = top_top_set_set_rule ) ) ).
% top.extremum_uniqueI
thf(fact_391_subset__UNIV,axiom,
! [A: set_rule] : ( ord_less_eq_set_rule @ A @ top_top_set_rule ) ).
% subset_UNIV
thf(fact_392_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_393_subset__UNIV,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ A @ top_top_set_rat ) ).
% subset_UNIV
thf(fact_394_subset__UNIV,axiom,
! [A: set_set_rat] : ( ord_le513522071413781156et_rat @ A @ top_top_set_set_rat ) ).
% subset_UNIV
thf(fact_395_subset__UNIV,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ top_top_set_set_nat ) ).
% subset_UNIV
thf(fact_396_subset__UNIV,axiom,
! [A: set_set_rule] : ( ord_le7968974978423766289t_rule @ A @ top_top_set_set_rule ) ).
% subset_UNIV
thf(fact_397_stream_Oset__intros_I2_J,axiom,
! [Y: rat,X22: stream_rat,X1: rat] :
( ( member_rat @ Y @ ( sset_rat @ X22 ) )
=> ( member_rat @ Y @ ( sset_rat @ ( sCons_rat @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_398_stream_Oset__intros_I2_J,axiom,
! [Y: nat,X22: stream_nat,X1: nat] :
( ( member_nat @ Y @ ( sset_nat @ X22 ) )
=> ( member_nat @ Y @ ( sset_nat @ ( sCons_nat @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_399_stream_Oset__intros_I2_J,axiom,
! [Y: rule,X22: stream_rule,X1: rule] :
( ( member_rule @ Y @ ( sset_rule @ X22 ) )
=> ( member_rule @ Y @ ( sset_rule @ ( sCons_rule @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_400_stream_Oset__intros_I1_J,axiom,
! [X1: rat,X22: stream_rat] : ( member_rat @ X1 @ ( sset_rat @ ( sCons_rat @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_401_stream_Oset__intros_I1_J,axiom,
! [X1: nat,X22: stream_nat] : ( member_nat @ X1 @ ( sset_nat @ ( sCons_nat @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_402_stream_Oset__intros_I1_J,axiom,
! [X1: rule,X22: stream_rule] : ( member_rule @ X1 @ ( sset_rule @ ( sCons_rule @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_403_stream_Oset__cases,axiom,
! [E: rat,A2: stream_rat] :
( ( member_rat @ E @ ( sset_rat @ A2 ) )
=> ( ! [Z22: stream_rat] :
( A2
!= ( sCons_rat @ E @ Z22 ) )
=> ~ ! [Z1: rat,Z22: stream_rat] :
( ( A2
= ( sCons_rat @ Z1 @ Z22 ) )
=> ~ ( member_rat @ E @ ( sset_rat @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_404_stream_Oset__cases,axiom,
! [E: nat,A2: stream_nat] :
( ( member_nat @ E @ ( sset_nat @ A2 ) )
=> ( ! [Z22: stream_nat] :
( A2
!= ( sCons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: stream_nat] :
( ( A2
= ( sCons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( sset_nat @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_405_stream_Oset__cases,axiom,
! [E: rule,A2: stream_rule] :
( ( member_rule @ E @ ( sset_rule @ A2 ) )
=> ( ! [Z22: stream_rule] :
( A2
!= ( sCons_rule @ E @ Z22 ) )
=> ~ ! [Z1: rule,Z22: stream_rule] :
( ( A2
= ( sCons_rule @ Z1 @ Z22 ) )
=> ~ ( member_rule @ E @ ( sset_rule @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_406_stream_Oset__induct,axiom,
! [X: rat,A2: stream_rat,P: rat > stream_rat > $o] :
( ( member_rat @ X @ ( sset_rat @ A2 ) )
=> ( ! [Z1: rat,Z22: stream_rat] : ( P @ Z1 @ ( sCons_rat @ Z1 @ Z22 ) )
=> ( ! [Z1: rat,Z22: stream_rat,Xa: rat] :
( ( member_rat @ Xa @ ( sset_rat @ Z22 ) )
=> ( ( P @ Xa @ Z22 )
=> ( P @ Xa @ ( sCons_rat @ Z1 @ Z22 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_407_stream_Oset__induct,axiom,
! [X: nat,A2: stream_nat,P: nat > stream_nat > $o] :
( ( member_nat @ X @ ( sset_nat @ A2 ) )
=> ( ! [Z1: nat,Z22: stream_nat] : ( P @ Z1 @ ( sCons_nat @ Z1 @ Z22 ) )
=> ( ! [Z1: nat,Z22: stream_nat,Xa: nat] :
( ( member_nat @ Xa @ ( sset_nat @ Z22 ) )
=> ( ( P @ Xa @ Z22 )
=> ( P @ Xa @ ( sCons_nat @ Z1 @ Z22 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_408_stream_Oset__induct,axiom,
! [X: rule,A2: stream_rule,P: rule > stream_rule > $o] :
( ( member_rule @ X @ ( sset_rule @ A2 ) )
=> ( ! [Z1: rule,Z22: stream_rule] : ( P @ Z1 @ ( sCons_rule @ Z1 @ Z22 ) )
=> ( ! [Z1: rule,Z22: stream_rule,Xa: rule] :
( ( member_rule @ Xa @ ( sset_rule @ Z22 ) )
=> ( ( P @ Xa @ Z22 )
=> ( P @ Xa @ ( sCons_rule @ Z1 @ Z22 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_409_Sup__upper2,axiom,
! [U: set_rat,A: set_set_rat,V: set_rat] :
( ( member_set_rat @ U @ A )
=> ( ( ord_less_eq_set_rat @ V @ U )
=> ( ord_less_eq_set_rat @ V @ ( comple3890839924845867745et_rat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_410_Sup__upper2,axiom,
! [U: set_nat,A: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_411_Sup__upper2,axiom,
! [U: set_rule,A: set_set_rule,V: set_rule] :
( ( member_set_rule @ U @ A )
=> ( ( ord_less_eq_set_rule @ V @ U )
=> ( ord_less_eq_set_rule @ V @ ( comple2146307154184993742t_rule @ A ) ) ) ) ).
% Sup_upper2
thf(fact_412_Sup__le__iff,axiom,
! [A: set_set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ B )
= ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
=> ( ord_less_eq_set_rat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_413_Sup__le__iff,axiom,
! [A: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ B )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_414_Sup__le__iff,axiom,
! [A: set_set_rule,B: set_rule] :
( ( ord_less_eq_set_rule @ ( comple2146307154184993742t_rule @ A ) @ B )
= ( ! [X3: set_rule] :
( ( member_set_rule @ X3 @ A )
=> ( ord_less_eq_set_rule @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_415_Sup__upper,axiom,
! [X: set_rat,A: set_set_rat] :
( ( member_set_rat @ X @ A )
=> ( ord_less_eq_set_rat @ X @ ( comple3890839924845867745et_rat @ A ) ) ) ).
% Sup_upper
thf(fact_416_Sup__upper,axiom,
! [X: set_nat,A: set_set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_417_Sup__upper,axiom,
! [X: set_rule,A: set_set_rule] :
( ( member_set_rule @ X @ A )
=> ( ord_less_eq_set_rule @ X @ ( comple2146307154184993742t_rule @ A ) ) ) ).
% Sup_upper
thf(fact_418_Sup__least,axiom,
! [A: set_set_rat,Z2: set_rat] :
( ! [X2: set_rat] :
( ( member_set_rat @ X2 @ A )
=> ( ord_less_eq_set_rat @ X2 @ Z2 ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_419_Sup__least,axiom,
! [A: set_set_nat,Z2: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_420_Sup__least,axiom,
! [A: set_set_rule,Z2: set_rule] :
( ! [X2: set_rule] :
( ( member_set_rule @ X2 @ A )
=> ( ord_less_eq_set_rule @ X2 @ Z2 ) )
=> ( ord_less_eq_set_rule @ ( comple2146307154184993742t_rule @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_421_Sup__mono,axiom,
! [A: set_set_rat,B2: set_set_rat] :
( ! [A3: set_rat] :
( ( member_set_rat @ A3 @ A )
=> ? [X6: set_rat] :
( ( member_set_rat @ X6 @ B2 )
& ( ord_less_eq_set_rat @ A3 @ X6 ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ ( comple3890839924845867745et_rat @ B2 ) ) ) ).
% Sup_mono
thf(fact_422_Sup__mono,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ A )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ B2 )
& ( ord_less_eq_set_nat @ A3 @ X6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_423_Sup__mono,axiom,
! [A: set_set_rule,B2: set_set_rule] :
( ! [A3: set_rule] :
( ( member_set_rule @ A3 @ A )
=> ? [X6: set_rule] :
( ( member_set_rule @ X6 @ B2 )
& ( ord_less_eq_set_rule @ A3 @ X6 ) ) )
=> ( ord_less_eq_set_rule @ ( comple2146307154184993742t_rule @ A ) @ ( comple2146307154184993742t_rule @ B2 ) ) ) ).
% Sup_mono
thf(fact_424_Sup__eqI,axiom,
! [A: set_set_rat,X: set_rat] :
( ! [Y5: set_rat] :
( ( member_set_rat @ Y5 @ A )
=> ( ord_less_eq_set_rat @ Y5 @ X ) )
=> ( ! [Y5: set_rat] :
( ! [Z4: set_rat] :
( ( member_set_rat @ Z4 @ A )
=> ( ord_less_eq_set_rat @ Z4 @ Y5 ) )
=> ( ord_less_eq_set_rat @ X @ Y5 ) )
=> ( ( comple3890839924845867745et_rat @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_425_Sup__eqI,axiom,
! [A: set_set_nat,X: set_nat] :
( ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ A )
=> ( ord_less_eq_set_nat @ Y5 @ X ) )
=> ( ! [Y5: set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ A )
=> ( ord_less_eq_set_nat @ Z4 @ Y5 ) )
=> ( ord_less_eq_set_nat @ X @ Y5 ) )
=> ( ( comple7399068483239264473et_nat @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_426_Sup__eqI,axiom,
! [A: set_set_rule,X: set_rule] :
( ! [Y5: set_rule] :
( ( member_set_rule @ Y5 @ A )
=> ( ord_less_eq_set_rule @ Y5 @ X ) )
=> ( ! [Y5: set_rule] :
( ! [Z4: set_rule] :
( ( member_set_rule @ Z4 @ A )
=> ( ord_less_eq_set_rule @ Z4 @ Y5 ) )
=> ( ord_less_eq_set_rule @ X @ Y5 ) )
=> ( ( comple2146307154184993742t_rule @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_427_not__UNIV__eq__Ici,axiom,
! [L2: rat] :
( top_top_set_rat
!= ( set_ord_atLeast_rat @ L2 ) ) ).
% not_UNIV_eq_Ici
thf(fact_428_stream_Opred__cong,axiom,
! [X: stream_rat,Ya: stream_rat,P: rat > $o,Pa: rat > $o] :
( ( X = Ya )
=> ( ! [Z: rat] :
( ( member_rat @ Z @ ( sset_rat @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( pred_stream_rat @ P @ X )
= ( pred_stream_rat @ Pa @ Ya ) ) ) ) ).
% stream.pred_cong
thf(fact_429_stream_Opred__cong,axiom,
! [X: stream_nat,Ya: stream_nat,P: nat > $o,Pa: nat > $o] :
( ( X = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( sset_nat @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( pred_stream_nat @ P @ X )
= ( pred_stream_nat @ Pa @ Ya ) ) ) ) ).
% stream.pred_cong
thf(fact_430_stream_Opred__cong,axiom,
! [X: stream_rule,Ya: stream_rule,P: rule > $o,Pa: rule > $o] :
( ( X = Ya )
=> ( ! [Z: rule] :
( ( member_rule @ Z @ ( sset_rule @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( pred_stream_rule @ P @ X )
= ( pred_stream_rule @ Pa @ Ya ) ) ) ) ).
% stream.pred_cong
thf(fact_431_fair__stream,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( fair_fair_rule @ ( fair_f4564919574533178778m_rule @ F ) ) ) ).
% fair_stream
thf(fact_432_fair__stream,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( fair_fair_nat @ ( fair_fair_stream_nat @ F ) ) ) ).
% fair_stream
thf(fact_433_fair__stream,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( fair_fair_rat @ ( fair_fair_stream_rat @ F ) ) ) ).
% fair_stream
thf(fact_434_fair__stream,axiom,
! [F: nat > set_rat] :
( ( ( image_nat_set_rat @ F @ top_top_set_nat )
= top_top_set_set_rat )
=> ( fair_fair_set_rat @ ( fair_f1980322424541659363et_rat @ F ) ) ) ).
% fair_stream
thf(fact_435_fair__stream,axiom,
! [F: nat > set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
=> ( fair_fair_set_nat @ ( fair_f5488550982935056091et_nat @ F ) ) ) ).
% fair_stream
thf(fact_436_fair__stream,axiom,
! [F: nat > set_rule] :
( ( ( image_nat_set_rule @ F @ top_top_set_nat )
= top_top_set_set_rule )
=> ( fair_fair_set_rule @ ( fair_f1814642869706786768t_rule @ F ) ) ) ).
% fair_stream
thf(fact_437_all__subset__image,axiom,
! [F: nat > rule,A: set_nat,P: set_rule > $o] :
( ( ! [B5: set_rule] :
( ( ord_less_eq_set_rule @ B5 @ ( image_nat_rule @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A )
=> ( P @ ( image_nat_rule @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_438_all__subset__image,axiom,
! [F: nat > rat,A: set_nat,P: set_rat > $o] :
( ( ! [B5: set_rat] :
( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A )
=> ( P @ ( image_nat_rat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_439_surjD,axiom,
! [F: rule > rule,Y: rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ? [X2: rule] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_440_surjD,axiom,
! [F: rule > nat,Y: nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ? [X2: rule] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_441_surjD,axiom,
! [F: rule > rat,Y: rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ? [X2: rule] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_442_surjD,axiom,
! [F: nat > rule,Y: rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_443_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_444_surjD,axiom,
! [F: nat > rat,Y: rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_445_surjD,axiom,
! [F: rat > rule,Y: rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ? [X2: rat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_446_surjD,axiom,
! [F: rat > nat,Y: nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ? [X2: rat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_447_surjD,axiom,
! [F: rat > rat,Y: rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ? [X2: rat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_448_surjD,axiom,
! [F: rule > set_rat,Y: set_rat] :
( ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat )
=> ? [X2: rule] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_449_surjE,axiom,
! [F: rule > rule,Y: rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ~ ! [X2: rule] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_450_surjE,axiom,
! [F: rule > nat,Y: nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ~ ! [X2: rule] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_451_surjE,axiom,
! [F: rule > rat,Y: rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ~ ! [X2: rule] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_452_surjE,axiom,
! [F: nat > rule,Y: rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_453_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_454_surjE,axiom,
! [F: nat > rat,Y: rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_455_surjE,axiom,
! [F: rat > rule,Y: rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ~ ! [X2: rat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_456_surjE,axiom,
! [F: rat > nat,Y: nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ~ ! [X2: rat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_457_surjE,axiom,
! [F: rat > rat,Y: rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ~ ! [X2: rat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_458_surjE,axiom,
! [F: rule > set_rat,Y: set_rat] :
( ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat )
=> ~ ! [X2: rule] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_459_surjI,axiom,
! [G: rule > rule,F: rule > rule] :
( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rule_rule @ G @ top_top_set_rule )
= top_top_set_rule ) ) ).
% surjI
thf(fact_460_surjI,axiom,
! [G: rule > nat,F: nat > rule] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rule_nat @ G @ top_top_set_rule )
= top_top_set_nat ) ) ).
% surjI
thf(fact_461_surjI,axiom,
! [G: rule > rat,F: rat > rule] :
( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rule_rat @ G @ top_top_set_rule )
= top_top_set_rat ) ) ).
% surjI
thf(fact_462_surjI,axiom,
! [G: nat > rule,F: rule > nat] :
( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_rule @ G @ top_top_set_nat )
= top_top_set_rule ) ) ).
% surjI
thf(fact_463_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_464_surjI,axiom,
! [G: nat > rat,F: rat > nat] :
( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_rat @ G @ top_top_set_nat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_465_surjI,axiom,
! [G: rat > rule,F: rule > rat] :
( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rat_rule @ G @ top_top_set_rat )
= top_top_set_rule ) ) ).
% surjI
thf(fact_466_surjI,axiom,
! [G: rat > nat,F: nat > rat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rat_nat @ G @ top_top_set_rat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_467_surjI,axiom,
! [G: rat > rat,F: rat > rat] :
( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rat_rat @ G @ top_top_set_rat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_468_surjI,axiom,
! [G: rule > set_rat,F: set_rat > rule] :
( ! [X2: set_rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rule_set_rat @ G @ top_top_set_rule )
= top_top_set_set_rat ) ) ).
% surjI
thf(fact_469_surj__def,axiom,
! [F: rule > rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
= ( ! [Y2: rule] :
? [X3: rule] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_470_surj__def,axiom,
! [F: rule > nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X3: rule] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_471_surj__def,axiom,
! [F: rule > rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
= ( ! [Y2: rat] :
? [X3: rule] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_472_surj__def,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
= ( ! [Y2: rule] :
? [X3: nat] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_473_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X3: nat] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_474_surj__def,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
= ( ! [Y2: rat] :
? [X3: nat] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_475_surj__def,axiom,
! [F: rat > rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
= ( ! [Y2: rule] :
? [X3: rat] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_476_surj__def,axiom,
! [F: rat > nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X3: rat] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_477_surj__def,axiom,
! [F: rat > rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
= ( ! [Y2: rat] :
? [X3: rat] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_478_surj__def,axiom,
! [F: rule > set_rat] :
( ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat )
= ( ! [Y2: set_rat] :
? [X3: rule] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_479_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_480_pinf_I6_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_481_pinf_I8_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_482_minf_I6_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_483_psubsetD,axiom,
! [A: set_rat,B2: set_rat,C2: rat] :
( ( ord_less_set_rat @ A @ B2 )
=> ( ( member_rat @ C2 @ A )
=> ( member_rat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_484_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_485_psubsetD,axiom,
! [A: set_rule,B2: set_rule,C2: rule] :
( ( ord_less_set_rule @ A @ B2 )
=> ( ( member_rule @ C2 @ A )
=> ( member_rule @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_486_Sup__SUP__eq,axiom,
( comple4580332206425622756_rat_o
= ( ^ [S3: set_rat_o,X3: rat] : ( member_rat @ X3 @ ( comple3890839924845867745et_rat @ ( image_rat_o_set_rat @ collect_rat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_487_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S3: set_nat_o,X3: nat] : ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_488_Sup__SUP__eq,axiom,
( comple1826244231481717815rule_o
= ( ^ [S3: set_rule_o,X3: rule] : ( member_rule @ X3 @ ( comple2146307154184993742t_rule @ ( image_1281159361656534528t_rule @ collect_rule @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_489_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_490_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_491_minf_I7_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ~ ( ord_less_nat @ T2 @ X6 ) ) ).
% minf(7)
thf(fact_492_minf_I5_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ord_less_nat @ X6 @ T2 ) ) ).
% minf(5)
thf(fact_493_minf_I4_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( X6 != T2 ) ) ).
% minf(4)
thf(fact_494_minf_I3_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( X6 != T2 ) ) ).
% minf(3)
thf(fact_495_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_496_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_497_pinf_I7_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ord_less_nat @ T2 @ X6 ) ) ).
% pinf(7)
thf(fact_498_pinf_I5_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ~ ( ord_less_nat @ X6 @ T2 ) ) ).
% pinf(5)
thf(fact_499_pinf_I4_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(4)
thf(fact_500_pinf_I3_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( X6 != T2 ) ) ).
% pinf(3)
thf(fact_501_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_502_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_503_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_504_minf_I8_J,axiom,
! [T2: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ~ ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_505_sset__smerge,axiom,
! [Ss: stream_stream_rat] :
( ( sset_rat @ ( smerge_rat @ Ss ) )
= ( comple3890839924845867745et_rat @ ( image_3934614161387701885et_rat @ sset_rat @ ( sset_stream_rat @ Ss ) ) ) ) ).
% sset_smerge
thf(fact_506_sset__smerge,axiom,
! [Ss: stream_stream_nat] :
( ( sset_nat @ ( smerge_nat @ Ss ) )
= ( comple7399068483239264473et_nat @ ( image_7912102293542740589et_nat @ sset_nat @ ( sset_stream_nat @ Ss ) ) ) ) ).
% sset_smerge
thf(fact_507_sset__smerge,axiom,
! [Ss: stream_stream_rule] :
( ( sset_rule @ ( smerge_rule @ Ss ) )
= ( comple2146307154184993742t_rule @ ( image_6459725099818367575t_rule @ sset_rule @ ( sset_stream_rule @ Ss ) ) ) ) ).
% sset_smerge
thf(fact_508_image__Fpow__mono,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ B2 )
=> ( ord_le7968974978423766289t_rule @ ( image_458447791132712456t_rule @ ( image_nat_rule @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_rule @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_509_image__Fpow__mono,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ B2 )
=> ( ord_le513522071413781156et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_rat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_510_top_Oordering__top__axioms,axiom,
orderi2038897200410189450t_rule @ ord_less_eq_set_rule @ ord_less_set_rule @ top_top_set_rule ).
% top.ordering_top_axioms
thf(fact_511_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_512_top_Oordering__top__axioms,axiom,
ordering_top_set_rat @ ord_less_eq_set_rat @ ord_less_set_rat @ top_top_set_rat ).
% top.ordering_top_axioms
thf(fact_513_top_Oordering__top__axioms,axiom,
orderi3870585680403650131et_rat @ ord_le513522071413781156et_rat @ ord_less_set_set_rat @ top_top_set_set_rat ).
% top.ordering_top_axioms
thf(fact_514_top_Oordering__top__axioms,axiom,
orderi1027199981026551883et_nat @ ord_le6893508408891458716et_nat @ ord_less_set_set_nat @ top_top_set_set_nat ).
% top.ordering_top_axioms
thf(fact_515_top_Oordering__top__axioms,axiom,
orderi4207796640154259904t_rule @ ord_le7968974978423766289t_rule @ ord_le3472952955546049285t_rule @ top_top_set_set_rule ).
% top.ordering_top_axioms
thf(fact_516_surj__from__nat,axiom,
( ( image_nat_nat @ from_nat_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_from_nat
thf(fact_517_surj__from__nat,axiom,
( ( image_nat_rat @ from_nat_rat @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_from_nat
thf(fact_518_surj__uminus,axiom,
( ( image_rat_rat @ uminus_uminus_rat @ top_top_set_rat )
= top_top_set_rat ) ).
% surj_uminus
thf(fact_519_surj__Compl__image__subset,axiom,
! [F: rule > rule,A: set_rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( ord_less_eq_set_rule @ ( uminus4869265918275750596t_rule @ ( image_rule_rule @ F @ A ) ) @ ( image_rule_rule @ F @ ( uminus4869265918275750596t_rule @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_520_surj__Compl__image__subset,axiom,
! [F: rule > nat,A: set_rule] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_rule_nat @ F @ A ) ) @ ( image_rule_nat @ F @ ( uminus4869265918275750596t_rule @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_521_surj__Compl__image__subset,axiom,
! [F: rule > rat,A: set_rule] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_rule_rat @ F @ A ) ) @ ( image_rule_rat @ F @ ( uminus4869265918275750596t_rule @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_522_surj__Compl__image__subset,axiom,
! [F: nat > rule,A: set_nat] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( ord_less_eq_set_rule @ ( uminus4869265918275750596t_rule @ ( image_nat_rule @ F @ A ) ) @ ( image_nat_rule @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_523_surj__Compl__image__subset,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_524_surj__Compl__image__subset,axiom,
! [F: nat > rat,A: set_nat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_nat_rat @ F @ A ) ) @ ( image_nat_rat @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_525_surj__Compl__image__subset,axiom,
! [F: rat > rule,A: set_rat] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ( ord_less_eq_set_rule @ ( uminus4869265918275750596t_rule @ ( image_rat_rule @ F @ A ) ) @ ( image_rat_rule @ F @ ( uminus2201863774496077783et_rat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_526_surj__Compl__image__subset,axiom,
! [F: rat > nat,A: set_rat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_rat_nat @ F @ A ) ) @ ( image_rat_nat @ F @ ( uminus2201863774496077783et_rat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_527_surj__Compl__image__subset,axiom,
! [F: rat > rat,A: set_rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_rat_rat @ F @ A ) ) @ ( image_rat_rat @ F @ ( uminus2201863774496077783et_rat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_528_surj__Compl__image__subset,axiom,
! [F: rule > set_rat,A: set_rule] :
( ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat )
=> ( ord_le513522071413781156et_rat @ ( uminus3456807040561714317et_rat @ ( image_rule_set_rat @ F @ A ) ) @ ( image_rule_set_rat @ F @ ( uminus4869265918275750596t_rule @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_529_rat__denum,axiom,
? [F2: nat > rat] :
( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat ) ).
% rat_denum
thf(fact_530_ComplI,axiom,
! [C2: rat,A: set_rat] :
( ~ ( member_rat @ C2 @ A )
=> ( member_rat @ C2 @ ( uminus2201863774496077783et_rat @ A ) ) ) ).
% ComplI
thf(fact_531_ComplI,axiom,
! [C2: nat,A: set_nat] :
( ~ ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_532_ComplI,axiom,
! [C2: rule,A: set_rule] :
( ~ ( member_rule @ C2 @ A )
=> ( member_rule @ C2 @ ( uminus4869265918275750596t_rule @ A ) ) ) ).
% ComplI
thf(fact_533_Compl__iff,axiom,
! [C2: rat,A: set_rat] :
( ( member_rat @ C2 @ ( uminus2201863774496077783et_rat @ A ) )
= ( ~ ( member_rat @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_534_Compl__iff,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_535_Compl__iff,axiom,
! [C2: rule,A: set_rule] :
( ( member_rule @ C2 @ ( uminus4869265918275750596t_rule @ A ) )
= ( ~ ( member_rule @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_536_ComplD,axiom,
! [C2: rat,A: set_rat] :
( ( member_rat @ C2 @ ( uminus2201863774496077783et_rat @ A ) )
=> ~ ( member_rat @ C2 @ A ) ) ).
% ComplD
thf(fact_537_ComplD,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C2 @ A ) ) ).
% ComplD
thf(fact_538_ComplD,axiom,
! [C2: rule,A: set_rule] :
( ( member_rule @ C2 @ ( uminus4869265918275750596t_rule @ A ) )
=> ~ ( member_rule @ C2 @ A ) ) ).
% ComplD
thf(fact_539_surj__nat__to__rat__surj,axiom,
( ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_nat_to_rat_surj
thf(fact_540_image__Pow__mono,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ B2 )
=> ( ord_le7968974978423766289t_rule @ ( image_458447791132712456t_rule @ ( image_nat_rule @ F ) @ ( pow_nat @ A ) ) @ ( pow_rule @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_541_image__Pow__mono,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ B2 )
=> ( ord_le513522071413781156et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) @ ( pow_nat @ A ) ) @ ( pow_rat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_542_inj__image__Compl__subset,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus4869265918275750596t_rule @ ( image_nat_rule @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_543_inj__image__Compl__subset,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus2201863774496077783et_rat @ ( image_nat_rat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_544_Union__Pow__eq,axiom,
! [A: set_rat] :
( ( comple3890839924845867745et_rat @ ( pow_rat @ A ) )
= A ) ).
% Union_Pow_eq
thf(fact_545_Union__Pow__eq,axiom,
! [A: set_nat] :
( ( comple7399068483239264473et_nat @ ( pow_nat @ A ) )
= A ) ).
% Union_Pow_eq
thf(fact_546_Union__Pow__eq,axiom,
! [A: set_rule] :
( ( comple2146307154184993742t_rule @ ( pow_rule @ A ) )
= A ) ).
% Union_Pow_eq
thf(fact_547_Pow__UNIV,axiom,
( ( pow_rule @ top_top_set_rule )
= top_top_set_set_rule ) ).
% Pow_UNIV
thf(fact_548_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_549_Pow__UNIV,axiom,
( ( pow_rat @ top_top_set_rat )
= top_top_set_set_rat ) ).
% Pow_UNIV
thf(fact_550_Pow__UNIV,axiom,
( ( pow_set_rat @ top_top_set_set_rat )
= top_to535976598521457322et_rat ) ).
% Pow_UNIV
thf(fact_551_Pow__UNIV,axiom,
( ( pow_set_nat @ top_top_set_set_nat )
= top_to1114752748106383522et_nat ) ).
% Pow_UNIV
thf(fact_552_Pow__UNIV,axiom,
( ( pow_set_rule @ top_top_set_set_rule )
= top_to1078124695766848407t_rule ) ).
% Pow_UNIV
thf(fact_553_inj__on__image__Pow,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( inj_on4755138273128556404t_rule @ ( image_nat_rule @ F ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_554_inj__on__image__Pow,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_555_inj__to__nat,axiom,
inj_on_nat_nat @ to_nat_nat @ top_top_set_nat ).
% inj_to_nat
thf(fact_556_inj__to__nat,axiom,
inj_on_rat_nat @ to_nat_rat @ top_top_set_rat ).
% inj_to_nat
thf(fact_557_inj__on__image,axiom,
! [F: nat > rule,A: set_set_nat] :
( ( inj_on_nat_rule @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on4755138273128556404t_rule @ ( image_nat_rule @ F ) @ A ) ) ).
% inj_on_image
thf(fact_558_inj__on__image,axiom,
! [F: nat > rat,A: set_set_nat] :
( ( inj_on_nat_rat @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_559_image__set__diff,axiom,
! [F: nat > rule,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( image_nat_rule @ F @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_set_rule @ ( image_nat_rule @ F @ A ) @ ( image_nat_rule @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_560_image__set__diff,axiom,
! [F: nat > rat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( image_nat_rat @ F @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_set_rat @ ( image_nat_rat @ F @ A ) @ ( image_nat_rat @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_561_inj__on__image__set__diff,axiom,
! [F: nat > rule,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rule @ F @ C )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( image_nat_rule @ F @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_set_rule @ ( image_nat_rule @ F @ A ) @ ( image_nat_rule @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_562_inj__on__image__set__diff,axiom,
! [F: nat > rat,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rat @ F @ C )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( image_nat_rat @ F @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_set_rat @ ( image_nat_rat @ F @ A ) @ ( image_nat_rat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_563_ex__inj,axiom,
? [To_nat: nat > nat] : ( inj_on_nat_nat @ To_nat @ top_top_set_nat ) ).
% ex_inj
thf(fact_564_ex__inj,axiom,
? [To_nat: rat > nat] : ( inj_on_rat_nat @ To_nat @ top_top_set_rat ) ).
% ex_inj
thf(fact_565_inj__on__image__Fpow,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( inj_on4755138273128556404t_rule @ ( image_nat_rule @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_566_inj__on__image__Fpow,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_567_psubset__imp__ex__mem,axiom,
! [A: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A @ B2 )
=> ? [B3: rat] : ( member_rat @ B3 @ ( minus_minus_set_rat @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_568_psubset__imp__ex__mem,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_569_psubset__imp__ex__mem,axiom,
! [A: set_rule,B2: set_rule] :
( ( ord_less_set_rule @ A @ B2 )
=> ? [B3: rule] : ( member_rule @ B3 @ ( minus_minus_set_rule @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_570_range__ex1__eq,axiom,
! [F: rule > rat,B: rat] :
( ( inj_on_rule_rat @ F @ top_top_set_rule )
=> ( ( member_rat @ B @ ( image_rule_rat @ F @ top_top_set_rule ) )
= ( ? [X3: rule] :
( ( B
= ( F @ X3 ) )
& ! [Y2: rule] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_571_range__ex1__eq,axiom,
! [F: rule > nat,B: nat] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ( member_nat @ B @ ( image_rule_nat @ F @ top_top_set_rule ) )
= ( ? [X3: rule] :
( ( B
= ( F @ X3 ) )
& ! [Y2: rule] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_572_range__ex1__eq,axiom,
! [F: rule > rule,B: rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ( member_rule @ B @ ( image_rule_rule @ F @ top_top_set_rule ) )
= ( ? [X3: rule] :
( ( B
= ( F @ X3 ) )
& ! [Y2: rule] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_573_range__ex1__eq,axiom,
! [F: nat > rat,B: rat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( member_rat @ B @ ( image_nat_rat @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y2: nat] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_574_range__ex1__eq,axiom,
! [F: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y2: nat] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_575_range__ex1__eq,axiom,
! [F: nat > rule,B: rule] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( member_rule @ B @ ( image_nat_rule @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y2: nat] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_576_range__ex1__eq,axiom,
! [F: rat > rat,B: rat] :
( ( inj_on_rat_rat @ F @ top_top_set_rat )
=> ( ( member_rat @ B @ ( image_rat_rat @ F @ top_top_set_rat ) )
= ( ? [X3: rat] :
( ( B
= ( F @ X3 ) )
& ! [Y2: rat] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_577_range__ex1__eq,axiom,
! [F: rat > nat,B: nat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( member_nat @ B @ ( image_rat_nat @ F @ top_top_set_rat ) )
= ( ? [X3: rat] :
( ( B
= ( F @ X3 ) )
& ! [Y2: rat] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_578_range__ex1__eq,axiom,
! [F: rat > rule,B: rule] :
( ( inj_on_rat_rule @ F @ top_top_set_rat )
=> ( ( member_rule @ B @ ( image_rat_rule @ F @ top_top_set_rat ) )
= ( ? [X3: rat] :
( ( B
= ( F @ X3 ) )
& ! [Y2: rat] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_579_range__ex1__eq,axiom,
! [F: set_rat > rat,B: rat] :
( ( inj_on_set_rat_rat @ F @ top_top_set_set_rat )
=> ( ( member_rat @ B @ ( image_set_rat_rat @ F @ top_top_set_set_rat ) )
= ( ? [X3: set_rat] :
( ( B
= ( F @ X3 ) )
& ! [Y2: set_rat] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_580_inj__image__eq__iff,axiom,
! [F: nat > rule,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( ( image_nat_rule @ F @ A )
= ( image_nat_rule @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_581_inj__image__eq__iff,axiom,
! [F: nat > rat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( ( image_nat_rat @ F @ A )
= ( image_nat_rat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_582_inj__image__mem__iff,axiom,
! [F: rule > rat,A2: rule,A: set_rule] :
( ( inj_on_rule_rat @ F @ top_top_set_rule )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_rule_rat @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_583_inj__image__mem__iff,axiom,
! [F: rule > nat,A2: rule,A: set_rule] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_rule_nat @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_584_inj__image__mem__iff,axiom,
! [F: rule > rule,A2: rule,A: set_rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_rule_rule @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_585_inj__image__mem__iff,axiom,
! [F: nat > rat,A2: nat,A: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_nat_rat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_586_inj__image__mem__iff,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_587_inj__image__mem__iff,axiom,
! [F: nat > rule,A2: nat,A: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_nat_rule @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_588_inj__image__mem__iff,axiom,
! [F: rat > rat,A2: rat,A: set_rat] :
( ( inj_on_rat_rat @ F @ top_top_set_rat )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_rat_rat @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_589_inj__image__mem__iff,axiom,
! [F: rat > nat,A2: rat,A: set_rat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_rat_nat @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_590_inj__image__mem__iff,axiom,
! [F: rat > rule,A2: rat,A: set_rat] :
( ( inj_on_rat_rule @ F @ top_top_set_rat )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_rat_rule @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_591_inj__image__mem__iff,axiom,
! [F: set_rat > rat,A2: set_rat,A: set_set_rat] :
( ( inj_on_set_rat_rat @ F @ top_top_set_set_rat )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_set_rat_rat @ F @ A ) )
= ( member_set_rat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_592_inj__on__image__mem__iff,axiom,
! [F: rat > rat,B2: set_rat,A2: rat,A: set_rat] :
( ( inj_on_rat_rat @ F @ B2 )
=> ( ( member_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_rat_rat @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_593_inj__on__image__mem__iff,axiom,
! [F: rat > nat,B2: set_rat,A2: rat,A: set_rat] :
( ( inj_on_rat_nat @ F @ B2 )
=> ( ( member_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_rat_nat @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_594_inj__on__image__mem__iff,axiom,
! [F: rat > rule,B2: set_rat,A2: rat,A: set_rat] :
( ( inj_on_rat_rule @ F @ B2 )
=> ( ( member_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_rat_rule @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_595_inj__on__image__mem__iff,axiom,
! [F: nat > rat,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_rat @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_nat_rat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_596_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_597_inj__on__image__mem__iff,axiom,
! [F: nat > rule,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_rule @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_nat_rule @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_598_inj__on__image__mem__iff,axiom,
! [F: rule > rat,B2: set_rule,A2: rule,A: set_rule] :
( ( inj_on_rule_rat @ F @ B2 )
=> ( ( member_rule @ A2 @ B2 )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_rule_rat @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_599_inj__on__image__mem__iff,axiom,
! [F: rule > nat,B2: set_rule,A2: rule,A: set_rule] :
( ( inj_on_rule_nat @ F @ B2 )
=> ( ( member_rule @ A2 @ B2 )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_rule_nat @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_600_inj__on__image__mem__iff,axiom,
! [F: rule > rule,B2: set_rule,A2: rule,A: set_rule] :
( ( inj_on_rule_rule @ F @ B2 )
=> ( ( member_rule @ A2 @ B2 )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_rule_rule @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_601_inj__on__image__eq__iff,axiom,
! [F: nat > rule,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rule @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_nat_rule @ F @ A )
= ( image_nat_rule @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_602_inj__on__image__eq__iff,axiom,
! [F: nat > rat,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_nat_rat @ F @ A )
= ( image_nat_rat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_603_image__strict__mono,axiom,
! [F: nat > rule,B2: set_nat,A: set_nat] :
( ( inj_on_nat_rule @ F @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_set_rule @ ( image_nat_rule @ F @ A ) @ ( image_nat_rule @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_604_image__strict__mono,axiom,
! [F: nat > rat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_rat @ F @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_set_rat @ ( image_nat_rat @ F @ A ) @ ( image_nat_rat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_605_image__diff__subset,axiom,
! [F: nat > rule,A: set_nat,B2: set_nat] : ( ord_less_eq_set_rule @ ( minus_minus_set_rule @ ( image_nat_rule @ F @ A ) @ ( image_nat_rule @ F @ B2 ) ) @ ( image_nat_rule @ F @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_606_image__diff__subset,axiom,
! [F: nat > rat,A: set_nat,B2: set_nat] : ( ord_less_eq_set_rat @ ( minus_minus_set_rat @ ( image_nat_rat @ F @ A ) @ ( image_nat_rat @ F @ B2 ) ) @ ( image_nat_rat @ F @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_607_Compl__eq__Diff__UNIV,axiom,
( uminus4869265918275750596t_rule
= ( minus_minus_set_rule @ top_top_set_rule ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_608_Compl__eq__Diff__UNIV,axiom,
( uminus5710092332889474511et_nat
= ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_609_Compl__eq__Diff__UNIV,axiom,
( uminus2201863774496077783et_rat
= ( minus_minus_set_rat @ top_top_set_rat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_610_Compl__eq__Diff__UNIV,axiom,
( uminus3456807040561714317et_rat
= ( minus_5007325069933123869et_rat @ top_top_set_set_rat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_611_Compl__eq__Diff__UNIV,axiom,
( uminus613421341184616069et_nat
= ( minus_2163939370556025621et_nat @ top_top_set_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_612_Compl__eq__Diff__UNIV,axiom,
( uminus4253708974368314874t_rule
= ( minus_5074010835065784970t_rule @ top_top_set_set_rule ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_613_subset__Pow__Union,axiom,
! [A: set_set_rat] : ( ord_le513522071413781156et_rat @ A @ ( pow_rat @ ( comple3890839924845867745et_rat @ A ) ) ) ).
% subset_Pow_Union
thf(fact_614_subset__Pow__Union,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( pow_nat @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% subset_Pow_Union
thf(fact_615_subset__Pow__Union,axiom,
! [A: set_set_rule] : ( ord_le7968974978423766289t_rule @ A @ ( pow_rule @ ( comple2146307154184993742t_rule @ A ) ) ) ).
% subset_Pow_Union
thf(fact_616_image__Pow__surj,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( ( image_nat_rule @ F @ A )
= B2 )
=> ( ( image_458447791132712456t_rule @ ( image_nat_rule @ F ) @ ( pow_nat @ A ) )
= ( pow_rule @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_617_image__Pow__surj,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( ( image_nat_rat @ F @ A )
= B2 )
=> ( ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) @ ( pow_nat @ A ) )
= ( pow_rat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_618_inj__image__subset__iff,axiom,
! [F: nat > rule,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ ( image_nat_rule @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_619_inj__image__subset__iff,axiom,
! [F: nat > rat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ ( image_nat_rat @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_620_range__diff,axiom,
! [A2: rat] :
( ( image_rat_rat @ ( minus_minus_rat @ A2 ) @ top_top_set_rat )
= top_top_set_rat ) ).
% range_diff
thf(fact_621_all__subset__image__inj,axiom,
! [F: nat > rule,S4: set_nat,P: set_rule > $o] :
( ( ! [T3: set_rule] :
( ( ord_less_eq_set_rule @ T3 @ ( image_nat_rule @ F @ S4 ) )
=> ( P @ T3 ) ) )
= ( ! [T3: set_nat] :
( ( ( ord_less_eq_set_nat @ T3 @ S4 )
& ( inj_on_nat_rule @ F @ T3 ) )
=> ( P @ ( image_nat_rule @ F @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_622_all__subset__image__inj,axiom,
! [F: nat > rat,S4: set_nat,P: set_rat > $o] :
( ( ! [T3: set_rat] :
( ( ord_less_eq_set_rat @ T3 @ ( image_nat_rat @ F @ S4 ) )
=> ( P @ T3 ) ) )
= ( ! [T3: set_nat] :
( ( ( ord_less_eq_set_nat @ T3 @ S4 )
& ( inj_on_nat_rat @ F @ T3 ) )
=> ( P @ ( image_nat_rat @ F @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_623_ex__subset__image__inj,axiom,
! [F: nat > rule,S4: set_nat,P: set_rule > $o] :
( ( ? [T3: set_rule] :
( ( ord_less_eq_set_rule @ T3 @ ( image_nat_rule @ F @ S4 ) )
& ( P @ T3 ) ) )
= ( ? [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S4 )
& ( inj_on_nat_rule @ F @ T3 )
& ( P @ ( image_nat_rule @ F @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_624_ex__subset__image__inj,axiom,
! [F: nat > rat,S4: set_nat,P: set_rat > $o] :
( ( ? [T3: set_rat] :
( ( ord_less_eq_set_rat @ T3 @ ( image_nat_rat @ F @ S4 ) )
& ( P @ T3 ) ) )
= ( ? [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S4 )
& ( inj_on_nat_rat @ F @ T3 )
& ( P @ ( image_nat_rat @ F @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_625_subset__image__inj,axiom,
! [S4: set_rule,F: nat > rule,T4: set_nat] :
( ( ord_less_eq_set_rule @ S4 @ ( image_nat_rule @ F @ T4 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T4 )
& ( inj_on_nat_rule @ F @ U2 )
& ( S4
= ( image_nat_rule @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_626_subset__image__inj,axiom,
! [S4: set_rat,F: nat > rat,T4: set_nat] :
( ( ord_less_eq_set_rat @ S4 @ ( image_nat_rat @ F @ T4 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T4 )
& ( inj_on_nat_rat @ F @ U2 )
& ( S4
= ( image_nat_rat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_627_Rats__eq__range__nat__to__rat__surj,axiom,
( field_6020823756834552118ts_rat
= ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat ) ) ).
% Rats_eq_range_nat_to_rat_surj
thf(fact_628_DiffI,axiom,
! [C2: rat,A: set_rat,B2: set_rat] :
( ( member_rat @ C2 @ A )
=> ( ~ ( member_rat @ C2 @ B2 )
=> ( member_rat @ C2 @ ( minus_minus_set_rat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_629_DiffI,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ~ ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_630_DiffI,axiom,
! [C2: rule,A: set_rule,B2: set_rule] :
( ( member_rule @ C2 @ A )
=> ( ~ ( member_rule @ C2 @ B2 )
=> ( member_rule @ C2 @ ( minus_minus_set_rule @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_631_Diff__iff,axiom,
! [C2: rat,A: set_rat,B2: set_rat] :
( ( member_rat @ C2 @ ( minus_minus_set_rat @ A @ B2 ) )
= ( ( member_rat @ C2 @ A )
& ~ ( member_rat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_632_Diff__iff,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat @ C2 @ A )
& ~ ( member_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_633_Diff__iff,axiom,
! [C2: rule,A: set_rule,B2: set_rule] :
( ( member_rule @ C2 @ ( minus_minus_set_rule @ A @ B2 ) )
= ( ( member_rule @ C2 @ A )
& ~ ( member_rule @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_634_DiffE,axiom,
! [C2: rat,A: set_rat,B2: set_rat] :
( ( member_rat @ C2 @ ( minus_minus_set_rat @ A @ B2 ) )
=> ~ ( ( member_rat @ C2 @ A )
=> ( member_rat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_635_DiffE,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_636_DiffE,axiom,
! [C2: rule,A: set_rule,B2: set_rule] :
( ( member_rule @ C2 @ ( minus_minus_set_rule @ A @ B2 ) )
=> ~ ( ( member_rule @ C2 @ A )
=> ( member_rule @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_637_DiffD1,axiom,
! [C2: rat,A: set_rat,B2: set_rat] :
( ( member_rat @ C2 @ ( minus_minus_set_rat @ A @ B2 ) )
=> ( member_rat @ C2 @ A ) ) ).
% DiffD1
thf(fact_638_DiffD1,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ( member_nat @ C2 @ A ) ) ).
% DiffD1
thf(fact_639_DiffD1,axiom,
! [C2: rule,A: set_rule,B2: set_rule] :
( ( member_rule @ C2 @ ( minus_minus_set_rule @ A @ B2 ) )
=> ( member_rule @ C2 @ A ) ) ).
% DiffD1
thf(fact_640_DiffD2,axiom,
! [C2: rat,A: set_rat,B2: set_rat] :
( ( member_rat @ C2 @ ( minus_minus_set_rat @ A @ B2 ) )
=> ~ ( member_rat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_641_DiffD2,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( member_nat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_642_DiffD2,axiom,
! [C2: rule,A: set_rule,B2: set_rule] :
( ( member_rule @ C2 @ ( minus_minus_set_rule @ A @ B2 ) )
=> ~ ( member_rule @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_643_Rats__minus__iff,axiom,
! [A2: rat] :
( ( member_rat @ ( uminus_uminus_rat @ A2 ) @ field_6020823756834552118ts_rat )
= ( member_rat @ A2 @ field_6020823756834552118ts_rat ) ) ).
% Rats_minus_iff
thf(fact_644_the__inv__into__into,axiom,
! [F: rat > rat,A: set_rat,X: rat,B2: set_rat] :
( ( inj_on_rat_rat @ F @ A )
=> ( ( member_rat @ X @ ( image_rat_rat @ F @ A ) )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( member_rat @ ( the_inv_into_rat_rat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_645_the__inv__into__into,axiom,
! [F: nat > rat,A: set_nat,X: rat,B2: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( member_rat @ X @ ( image_nat_rat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_inv_into_nat_rat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_646_the__inv__into__into,axiom,
! [F: rule > rat,A: set_rule,X: rat,B2: set_rule] :
( ( inj_on_rule_rat @ F @ A )
=> ( ( member_rat @ X @ ( image_rule_rat @ F @ A ) )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( member_rule @ ( the_in8146750563547750866le_rat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_647_the__inv__into__into,axiom,
! [F: rat > nat,A: set_rat,X: nat,B2: set_rat] :
( ( inj_on_rat_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_rat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( member_rat @ ( the_inv_into_rat_nat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_648_the__inv__into__into,axiom,
! [F: nat > nat,A: set_nat,X: nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_649_the__inv__into__into,axiom,
! [F: rule > nat,A: set_rule,X: nat,B2: set_rule] :
( ( inj_on_rule_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_rule_nat @ F @ A ) )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( member_rule @ ( the_in8781880623634246602le_nat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_650_the__inv__into__into,axiom,
! [F: rat > rule,A: set_rat,X: rule,B2: set_rat] :
( ( inj_on_rat_rule @ F @ A )
=> ( ( member_rule @ X @ ( image_rat_rule @ F @ A ) )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( member_rat @ ( the_in2217467737105579218t_rule @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_651_the__inv__into__into,axiom,
! [F: nat > rule,A: set_nat,X: rule,B2: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( member_rule @ X @ ( image_nat_rule @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_in5544616208814386890t_rule @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_652_the__inv__into__into,axiom,
! [F: rule > rule,A: set_rule,X: rule,B2: set_rule] :
( ( inj_on_rule_rule @ F @ A )
=> ( ( member_rule @ X @ ( image_rule_rule @ F @ A ) )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( member_rule @ ( the_in80044576880915775e_rule @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_653_the__inv__into__onto,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( image_rule_nat @ ( the_in5544616208814386890t_rule @ A @ F ) @ ( image_nat_rule @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_654_the__inv__into__onto,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( image_rat_nat @ ( the_inv_into_nat_rat @ A @ F ) @ ( image_nat_rat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_655_the__inv__into__onto,axiom,
! [F: rule > nat,A: set_rule] :
( ( inj_on_rule_nat @ F @ A )
=> ( ( image_nat_rule @ ( the_in8781880623634246602le_nat @ A @ F ) @ ( image_rule_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_656_the__inv__into__onto,axiom,
! [F: rat > nat,A: set_rat] :
( ( inj_on_rat_nat @ F @ A )
=> ( ( image_nat_rat @ ( the_inv_into_rat_nat @ A @ F ) @ ( image_rat_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_657_from__nat__def,axiom,
( from_nat_nat
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ to_nat_nat ) ) ).
% from_nat_def
thf(fact_658_from__nat__def,axiom,
( from_nat_rat
= ( hilber3317322552863949046at_nat @ top_top_set_rat @ to_nat_rat ) ) ).
% from_nat_def
thf(fact_659_inv__into__image__cancel,axiom,
! [F: nat > rule,A: set_nat,S4: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( ord_less_eq_set_nat @ S4 @ A )
=> ( ( image_rule_nat @ ( hilber8541579349336805475t_rule @ A @ F ) @ ( image_nat_rule @ F @ S4 ) )
= S4 ) ) ) ).
% inv_into_image_cancel
thf(fact_660_inv__into__image__cancel,axiom,
! [F: nat > rat,A: set_nat,S4: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( ord_less_eq_set_nat @ S4 @ A )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ A @ F ) @ ( image_nat_rat @ F @ S4 ) )
= S4 ) ) ) ).
% inv_into_image_cancel
thf(fact_661_inv__into__image__cancel,axiom,
! [F: rule > nat,A: set_rule,S4: set_rule] :
( ( inj_on_rule_nat @ F @ A )
=> ( ( ord_less_eq_set_rule @ S4 @ A )
=> ( ( image_nat_rule @ ( hilber2555471727301889379le_nat @ A @ F ) @ ( image_rule_nat @ F @ S4 ) )
= S4 ) ) ) ).
% inv_into_image_cancel
thf(fact_662_inv__into__image__cancel,axiom,
! [F: rat > nat,A: set_rat,S4: set_rat] :
( ( inj_on_rat_nat @ F @ A )
=> ( ( ord_less_eq_set_rat @ S4 @ A )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ A @ F ) @ ( image_rat_nat @ F @ S4 ) )
= S4 ) ) ) ).
% inv_into_image_cancel
thf(fact_663_inv__into__injective,axiom,
! [A: set_nat,F: nat > rat,X: rat,Y: rat] :
( ( ( hilber2998747136712319222at_rat @ A @ F @ X )
= ( hilber2998747136712319222at_rat @ A @ F @ Y ) )
=> ( ( member_rat @ X @ ( image_nat_rat @ F @ A ) )
=> ( ( member_rat @ Y @ ( image_nat_rat @ F @ A ) )
=> ( X = Y ) ) ) ) ).
% inv_into_injective
thf(fact_664_inv__into__injective,axiom,
! [A: set_nat,F: nat > rule,X: rule,Y: rule] :
( ( ( hilber8541579349336805475t_rule @ A @ F @ X )
= ( hilber8541579349336805475t_rule @ A @ F @ Y ) )
=> ( ( member_rule @ X @ ( image_nat_rule @ F @ A ) )
=> ( ( member_rule @ Y @ ( image_nat_rule @ F @ A ) )
=> ( X = Y ) ) ) ) ).
% inv_into_injective
thf(fact_665_inv__into__into,axiom,
! [X: rat,F: rat > rat,A: set_rat] :
( ( member_rat @ X @ ( image_rat_rat @ F @ A ) )
=> ( member_rat @ ( hilber2682192492777453310at_rat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_666_inv__into__into,axiom,
! [X: rat,F: nat > rat,A: set_nat] :
( ( member_rat @ X @ ( image_nat_rat @ F @ A ) )
=> ( member_nat @ ( hilber2998747136712319222at_rat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_667_inv__into__into,axiom,
! [X: rat,F: rule > rat,A: set_rule] :
( ( member_rat @ X @ ( image_rule_rat @ F @ A ) )
=> ( member_rule @ ( hilber1920341667215393643le_rat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_668_inv__into__into,axiom,
! [X: nat,F: rat > nat,A: set_rat] :
( ( member_nat @ X @ ( image_rat_nat @ F @ A ) )
=> ( member_rat @ ( hilber3317322552863949046at_nat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_669_inv__into__into,axiom,
! [X: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( member_nat @ ( hilber3633877196798814958at_nat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_670_inv__into__into,axiom,
! [X: nat,F: rule > nat,A: set_rule] :
( ( member_nat @ X @ ( image_rule_nat @ F @ A ) )
=> ( member_rule @ ( hilber2555471727301889379le_nat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_671_inv__into__into,axiom,
! [X: rule,F: rat > rule,A: set_rat] :
( ( member_rule @ X @ ( image_rat_rule @ F @ A ) )
=> ( member_rat @ ( hilber5214430877627997803t_rule @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_672_inv__into__into,axiom,
! [X: rule,F: nat > rule,A: set_nat] :
( ( member_rule @ X @ ( image_nat_rule @ F @ A ) )
=> ( member_nat @ ( hilber8541579349336805475t_rule @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_673_inv__into__into,axiom,
! [X: rule,F: rule > rule,A: set_rule] :
( ( member_rule @ X @ ( image_rule_rule @ F @ A ) )
=> ( member_rule @ ( hilber2978553400015838680e_rule @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_674_f__inv__into__f,axiom,
! [Y: rat,F: nat > rat,A: set_nat] :
( ( member_rat @ Y @ ( image_nat_rat @ F @ A ) )
=> ( ( F @ ( hilber2998747136712319222at_rat @ A @ F @ Y ) )
= Y ) ) ).
% f_inv_into_f
thf(fact_675_f__inv__into__f,axiom,
! [Y: rule,F: nat > rule,A: set_nat] :
( ( member_rule @ Y @ ( image_nat_rule @ F @ A ) )
=> ( ( F @ ( hilber8541579349336805475t_rule @ A @ F @ Y ) )
= Y ) ) ).
% f_inv_into_f
thf(fact_676_surj__f__inv__f,axiom,
! [F: rule > rule,Y: rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( ( F @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_677_surj__f__inv__f,axiom,
! [F: rule > nat,Y: nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( ( F @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_678_surj__f__inv__f,axiom,
! [F: rule > rat,Y: rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( ( F @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_679_surj__f__inv__f,axiom,
! [F: nat > rule,Y: rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( ( F @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_680_surj__f__inv__f,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_681_surj__f__inv__f,axiom,
! [F: nat > rat,Y: rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ( F @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_682_surj__f__inv__f,axiom,
! [F: rat > rule,Y: rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ( ( F @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_683_surj__f__inv__f,axiom,
! [F: rat > nat,Y: nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ( F @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_684_surj__f__inv__f,axiom,
! [F: rat > rat,Y: rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ( F @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_685_surj__f__inv__f,axiom,
! [F: rule > set_rat,Y: set_rat] :
( ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat )
=> ( ( F @ ( hilber7153471463942364961et_rat @ top_top_set_rule @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_686_surj__iff__all,axiom,
! [F: rule > rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
= ( ! [X3: rule] :
( ( F @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_687_surj__iff__all,axiom,
! [F: rule > nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
= ( ! [X3: nat] :
( ( F @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_688_surj__iff__all,axiom,
! [F: rule > rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
= ( ! [X3: rat] :
( ( F @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_689_surj__iff__all,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
= ( ! [X3: rule] :
( ( F @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_690_surj__iff__all,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [X3: nat] :
( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_691_surj__iff__all,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
= ( ! [X3: rat] :
( ( F @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_692_surj__iff__all,axiom,
! [F: rat > rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
= ( ! [X3: rule] :
( ( F @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_693_surj__iff__all,axiom,
! [F: rat > nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
= ( ! [X3: nat] :
( ( F @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_694_surj__iff__all,axiom,
! [F: rat > rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
= ( ! [X3: rat] :
( ( F @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_695_surj__iff__all,axiom,
! [F: rule > set_rat] :
( ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat )
= ( ! [X3: set_rat] :
( ( F @ ( hilber7153471463942364961et_rat @ top_top_set_rule @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_696_image__f__inv__f,axiom,
! [F: rule > rule,A: set_rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( ( image_rule_rule @ F @ ( image_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_697_image__f__inv__f,axiom,
! [F: rule > nat,A: set_nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( ( image_rule_nat @ F @ ( image_nat_rule @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_698_image__f__inv__f,axiom,
! [F: rule > rat,A: set_rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( ( image_rule_rat @ F @ ( image_rat_rule @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_699_image__f__inv__f,axiom,
! [F: nat > rule,A: set_rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( ( image_nat_rule @ F @ ( image_rule_nat @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_700_image__f__inv__f,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_701_image__f__inv__f,axiom,
! [F: nat > rat,A: set_rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_nat_rat @ F @ ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_702_image__f__inv__f,axiom,
! [F: rat > rule,A: set_rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ( ( image_rat_rule @ F @ ( image_rule_rat @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_703_image__f__inv__f,axiom,
! [F: rat > nat,A: set_nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_rat_nat @ F @ ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_704_image__f__inv__f,axiom,
! [F: rat > rat,A: set_rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_rat_rat @ F @ ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_705_image__f__inv__f,axiom,
! [F: rule > set_rat,A: set_set_rat] :
( ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat )
=> ( ( image_rule_set_rat @ F @ ( image_set_rat_rule @ ( hilber7153471463942364961et_rat @ top_top_set_rule @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_706_surj__imp__inv__eq,axiom,
! [F: rule > rule,G: rule > rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber2978553400015838680e_rule @ top_top_set_rule @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_707_surj__imp__inv__eq,axiom,
! [F: rule > nat,G: nat > rule] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber2555471727301889379le_nat @ top_top_set_rule @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_708_surj__imp__inv__eq,axiom,
! [F: rule > rat,G: rat > rule] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber1920341667215393643le_rat @ top_top_set_rule @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_709_surj__imp__inv__eq,axiom,
! [F: nat > rule,G: rule > nat] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber8541579349336805475t_rule @ top_top_set_nat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_710_surj__imp__inv__eq,axiom,
! [F: nat > nat,G: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_711_surj__imp__inv__eq,axiom,
! [F: nat > rat,G: rat > nat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber2998747136712319222at_rat @ top_top_set_nat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_712_surj__imp__inv__eq,axiom,
! [F: rat > rule,G: rule > rat] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber5214430877627997803t_rule @ top_top_set_rat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_713_surj__imp__inv__eq,axiom,
! [F: rat > nat,G: nat > rat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber3317322552863949046at_nat @ top_top_set_rat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_714_surj__imp__inv__eq,axiom,
! [F: rat > rat,G: rat > rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_715_surj__imp__inv__eq,axiom,
! [F: rule > set_rat,G: set_rat > rule] :
( ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat )
=> ( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber7153471463942364961et_rat @ top_top_set_rule @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_716_image__inv__into__cancel,axiom,
! [F: rule > nat,A: set_rule,A7: set_nat,B7: set_nat] :
( ( ( image_rule_nat @ F @ A )
= A7 )
=> ( ( ord_less_eq_set_nat @ B7 @ A7 )
=> ( ( image_rule_nat @ F @ ( image_nat_rule @ ( hilber2555471727301889379le_nat @ A @ F ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_717_image__inv__into__cancel,axiom,
! [F: rat > nat,A: set_rat,A7: set_nat,B7: set_nat] :
( ( ( image_rat_nat @ F @ A )
= A7 )
=> ( ( ord_less_eq_set_nat @ B7 @ A7 )
=> ( ( image_rat_nat @ F @ ( image_nat_rat @ ( hilber3317322552863949046at_nat @ A @ F ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_718_image__inv__into__cancel,axiom,
! [F: nat > rule,A: set_nat,A7: set_rule,B7: set_rule] :
( ( ( image_nat_rule @ F @ A )
= A7 )
=> ( ( ord_less_eq_set_rule @ B7 @ A7 )
=> ( ( image_nat_rule @ F @ ( image_rule_nat @ ( hilber8541579349336805475t_rule @ A @ F ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_719_image__inv__into__cancel,axiom,
! [F: nat > rat,A: set_nat,A7: set_rat,B7: set_rat] :
( ( ( image_nat_rat @ F @ A )
= A7 )
=> ( ( ord_less_eq_set_rat @ B7 @ A7 )
=> ( ( image_nat_rat @ F @ ( image_rat_nat @ ( hilber2998747136712319222at_rat @ A @ F ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_720_surj__imp__inj__inv,axiom,
! [F: rule > rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( inj_on_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ top_top_set_rule ) ) ).
% surj_imp_inj_inv
thf(fact_721_surj__imp__inj__inv,axiom,
! [F: rule > nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( inj_on_nat_rule @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_722_surj__imp__inj__inv,axiom,
! [F: rule > rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( inj_on_rat_rule @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F ) @ top_top_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_723_surj__imp__inj__inv,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( inj_on_rule_nat @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) @ top_top_set_rule ) ) ).
% surj_imp_inj_inv
thf(fact_724_surj__imp__inj__inv,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_725_surj__imp__inj__inv,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( inj_on_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) @ top_top_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_726_surj__imp__inj__inv,axiom,
! [F: rat > rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ( inj_on_rule_rat @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F ) @ top_top_set_rule ) ) ).
% surj_imp_inj_inv
thf(fact_727_surj__imp__inj__inv,axiom,
! [F: rat > nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( inj_on_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_728_surj__imp__inj__inv,axiom,
! [F: rat > rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( inj_on_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ top_top_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_729_surj__imp__inj__inv,axiom,
! [F: rule > set_rat] :
( ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat )
=> ( inj_on_set_rat_rule @ ( hilber7153471463942364961et_rat @ top_top_set_rule @ F ) @ top_top_set_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_730_inj__imp__surj__inv,axiom,
! [F: rule > rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ( image_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ top_top_set_rule )
= top_top_set_rule ) ) ).
% inj_imp_surj_inv
thf(fact_731_inj__imp__surj__inv,axiom,
! [F: rule > nat] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ( image_nat_rule @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F ) @ top_top_set_nat )
= top_top_set_rule ) ) ).
% inj_imp_surj_inv
thf(fact_732_inj__imp__surj__inv,axiom,
! [F: rule > rat] :
( ( inj_on_rule_rat @ F @ top_top_set_rule )
=> ( ( image_rat_rule @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F ) @ top_top_set_rat )
= top_top_set_rule ) ) ).
% inj_imp_surj_inv
thf(fact_733_inj__imp__surj__inv,axiom,
! [F: nat > rule] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( image_rule_nat @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) @ top_top_set_rule )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_734_inj__imp__surj__inv,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_735_inj__imp__surj__inv,axiom,
! [F: nat > rat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) @ top_top_set_rat )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_736_inj__imp__surj__inv,axiom,
! [F: rat > rule] :
( ( inj_on_rat_rule @ F @ top_top_set_rat )
=> ( ( image_rule_rat @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F ) @ top_top_set_rule )
= top_top_set_rat ) ) ).
% inj_imp_surj_inv
thf(fact_737_inj__imp__surj__inv,axiom,
! [F: rat > nat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) @ top_top_set_nat )
= top_top_set_rat ) ) ).
% inj_imp_surj_inv
thf(fact_738_inj__imp__surj__inv,axiom,
! [F: rat > rat] :
( ( inj_on_rat_rat @ F @ top_top_set_rat )
=> ( ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ top_top_set_rat )
= top_top_set_rat ) ) ).
% inj_imp_surj_inv
thf(fact_739_inj__imp__surj__inv,axiom,
! [F: rule > set_rat] :
( ( inj_on_rule_set_rat @ F @ top_top_set_rule )
=> ( ( image_set_rat_rule @ ( hilber7153471463942364961et_rat @ top_top_set_rule @ F ) @ top_top_set_set_rat )
= top_top_set_rule ) ) ).
% inj_imp_surj_inv
thf(fact_740_image__inv__f__f,axiom,
! [F: rule > nat,A: set_rule] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ( image_nat_rule @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F ) @ ( image_rule_nat @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_741_image__inv__f__f,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( image_rule_nat @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) @ ( image_nat_rule @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_742_image__inv__f__f,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) @ ( image_nat_rat @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_743_image__inv__f__f,axiom,
! [F: rat > nat,A: set_rat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) @ ( image_rat_nat @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_744_inj__transfer,axiom,
! [F: rule > rat,P: rule > $o,X: rule] :
( ( inj_on_rule_rat @ F @ top_top_set_rule )
=> ( ! [Y5: rat] :
( ( member_rat @ Y5 @ ( image_rule_rat @ F @ top_top_set_rule ) )
=> ( P @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_745_inj__transfer,axiom,
! [F: rule > nat,P: rule > $o,X: rule] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ ( image_rule_nat @ F @ top_top_set_rule ) )
=> ( P @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_746_inj__transfer,axiom,
! [F: rule > rule,P: rule > $o,X: rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ! [Y5: rule] :
( ( member_rule @ Y5 @ ( image_rule_rule @ F @ top_top_set_rule ) )
=> ( P @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_747_inj__transfer,axiom,
! [F: nat > rat,P: nat > $o,X: nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ! [Y5: rat] :
( ( member_rat @ Y5 @ ( image_nat_rat @ F @ top_top_set_nat ) )
=> ( P @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_748_inj__transfer,axiom,
! [F: nat > nat,P: nat > $o,X: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( P @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_749_inj__transfer,axiom,
! [F: nat > rule,P: nat > $o,X: nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ! [Y5: rule] :
( ( member_rule @ Y5 @ ( image_nat_rule @ F @ top_top_set_nat ) )
=> ( P @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_750_inj__transfer,axiom,
! [F: rat > rat,P: rat > $o,X: rat] :
( ( inj_on_rat_rat @ F @ top_top_set_rat )
=> ( ! [Y5: rat] :
( ( member_rat @ Y5 @ ( image_rat_rat @ F @ top_top_set_rat ) )
=> ( P @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_751_inj__transfer,axiom,
! [F: rat > nat,P: rat > $o,X: rat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ ( image_rat_nat @ F @ top_top_set_rat ) )
=> ( P @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_752_inj__transfer,axiom,
! [F: rat > rule,P: rat > $o,X: rat] :
( ( inj_on_rat_rule @ F @ top_top_set_rat )
=> ( ! [Y5: rule] :
( ( member_rule @ Y5 @ ( image_rat_rule @ F @ top_top_set_rat ) )
=> ( P @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_753_inj__transfer,axiom,
! [F: set_rat > rat,P: set_rat > $o,X: set_rat] :
( ( inj_on_set_rat_rat @ F @ top_top_set_set_rat )
=> ( ! [Y5: rat] :
( ( member_rat @ Y5 @ ( image_set_rat_rat @ F @ top_top_set_set_rat ) )
=> ( P @ ( hilber7959523950909280948at_rat @ top_top_set_set_rat @ F @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_754_inj__on__inv__into,axiom,
! [B2: set_rule,F: nat > rule,A: set_nat] :
( ( ord_less_eq_set_rule @ B2 @ ( image_nat_rule @ F @ A ) )
=> ( inj_on_rule_nat @ ( hilber8541579349336805475t_rule @ A @ F ) @ B2 ) ) ).
% inj_on_inv_into
thf(fact_755_inj__on__inv__into,axiom,
! [B2: set_rat,F: nat > rat,A: set_nat] :
( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F @ A ) )
=> ( inj_on_rat_nat @ ( hilber2998747136712319222at_rat @ A @ F ) @ B2 ) ) ).
% inj_on_inv_into
thf(fact_756_inj__on__the__inv__into,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( inj_on_rule_nat @ ( the_in5544616208814386890t_rule @ A @ F ) @ ( image_nat_rule @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_757_inj__on__the__inv__into,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( inj_on_rat_nat @ ( the_inv_into_nat_rat @ A @ F ) @ ( image_nat_rat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_758_f__the__inv__into__f,axiom,
! [F: nat > rat,A: set_nat,Y: rat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( member_rat @ Y @ ( image_nat_rat @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_rat @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_759_f__the__inv__into__f,axiom,
! [F: nat > rule,A: set_nat,Y: rule] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( member_rule @ Y @ ( image_nat_rule @ F @ A ) )
=> ( ( F @ ( the_in5544616208814386890t_rule @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_760_bijection_Oinj__inv,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( inj_on_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ top_top_set_rule ) ) ).
% bijection.inj_inv
thf(fact_761_bijection_Oinj__inv,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat ) ) ).
% bijection.inj_inv
thf(fact_762_bijection_Oinj__inv,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( inj_on_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ top_top_set_rat ) ) ).
% bijection.inj_inv
thf(fact_763_bijection_Oinj__inv,axiom,
! [F: set_rat > set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( inj_on626919071704544911et_rat @ ( hilber3461591988630719082et_rat @ top_top_set_set_rat @ F ) @ top_top_set_set_rat ) ) ).
% bijection.inj_inv
thf(fact_764_bijection_Oinj__inv,axiom,
! [F: set_nat > set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( inj_on4604407203859583615et_nat @ ( hilber7439080120785757786et_nat @ top_top_set_set_nat @ F ) @ top_top_set_set_nat ) ) ).
% bijection.inj_inv
thf(fact_765_bijection_Oinj__inv,axiom,
! [F: set_rule > set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( inj_on733622247606601321t_rule @ ( hilber824017158074262596t_rule @ top_top_set_set_rule @ F ) @ top_top_set_set_rule ) ) ).
% bijection.inj_inv
thf(fact_766_bijection_Osurj__inv,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( image_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ top_top_set_rule )
= top_top_set_rule ) ) ).
% bijection.surj_inv
thf(fact_767_bijection_Osurj__inv,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj_inv
thf(fact_768_bijection_Osurj__inv,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ top_top_set_rat )
= top_top_set_rat ) ) ).
% bijection.surj_inv
thf(fact_769_bijection_Osurj__inv,axiom,
! [F: set_rat > set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( ( image_3939399684171694371et_rat @ ( hilber3461591988630719082et_rat @ top_top_set_set_rat @ F ) @ top_top_set_set_rat )
= top_top_set_set_rat ) ) ).
% bijection.surj_inv
thf(fact_770_bijection_Osurj__inv,axiom,
! [F: set_nat > set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( ( image_7916887816326733075et_nat @ ( hilber7439080120785757786et_nat @ top_top_set_set_nat @ F ) @ top_top_set_set_nat )
= top_top_set_set_nat ) ) ).
% bijection.surj_inv
thf(fact_771_bijection_Osurj__inv,axiom,
! [F: set_rule > set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( ( image_2455769455774476541t_rule @ ( hilber824017158074262596t_rule @ top_top_set_set_rule @ F ) @ top_top_set_set_rule )
= top_top_set_set_rule ) ) ).
% bijection.surj_inv
thf(fact_772_inj__imp__bij__betw__inv,axiom,
! [F: nat > rule,M2: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( bij_betw_rule_nat @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) @ ( image_nat_rule @ F @ M2 ) @ M2 ) ) ).
% inj_imp_bij_betw_inv
thf(fact_773_inj__imp__bij__betw__inv,axiom,
! [F: nat > rat,M2: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( bij_betw_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) @ ( image_nat_rat @ F @ M2 ) @ M2 ) ) ).
% inj_imp_bij_betw_inv
thf(fact_774_strict__mono__inv__on__range,axiom,
! [F: rat > nat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( monotone_on_nat_rat @ ( image_rat_nat @ F @ top_top_set_rat ) @ ord_less_nat @ ord_less_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) ) ) ).
% strict_mono_inv_on_range
thf(fact_775_strict__mono__inv__on__range,axiom,
! [F: nat > rat] :
( ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_nat @ ord_less_rat @ F )
=> ( monotone_on_rat_nat @ ( image_nat_rat @ F @ top_top_set_nat ) @ ord_less_rat @ ord_less_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) ) ) ).
% strict_mono_inv_on_range
thf(fact_776_strict__mono__inv__on__range,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ ord_less_nat @ ord_less_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) ) ) ).
% strict_mono_inv_on_range
thf(fact_777_Sup__greaterThanAtLeast,axiom,
! [X: set_set_rat] :
( ( ord_less_set_set_rat @ X @ top_top_set_set_rat )
=> ( ( comple3392050375588816791et_rat @ ( set_or6674600949247491550et_rat @ X ) )
= top_top_set_set_rat ) ) ).
% Sup_greaterThanAtLeast
thf(fact_778_Sup__greaterThanAtLeast,axiom,
! [X: set_set_nat] :
( ( ord_less_set_set_nat @ X @ top_top_set_set_nat )
=> ( ( comple548664676211718543et_nat @ ( set_or3831215249870393302et_nat @ X ) )
= top_top_set_set_nat ) ) ).
% Sup_greaterThanAtLeast
thf(fact_779_Sup__greaterThanAtLeast,axiom,
! [X: set_set_rule] :
( ( ord_le3472952955546049285t_rule @ X @ top_top_set_set_rule )
=> ( ( comple4235951075906658052t_rule @ ( set_or6920019677468954187t_rule @ X ) )
= top_top_set_set_rule ) ) ).
% Sup_greaterThanAtLeast
thf(fact_780_Sup__greaterThanAtLeast,axiom,
! [X: set_rat] :
( ( ord_less_set_rat @ X @ top_top_set_rat )
=> ( ( comple3890839924845867745et_rat @ ( set_or6174011595382531368et_rat @ X ) )
= top_top_set_rat ) ) ).
% Sup_greaterThanAtLeast
thf(fact_781_Sup__greaterThanAtLeast,axiom,
! [X: set_nat] :
( ( ord_less_set_nat @ X @ top_top_set_nat )
=> ( ( comple7399068483239264473et_nat @ ( set_or458868116921152288et_nat @ X ) )
= top_top_set_nat ) ) ).
% Sup_greaterThanAtLeast
thf(fact_782_Sup__greaterThanAtLeast,axiom,
! [X: set_rule] :
( ( ord_less_set_rule @ X @ top_top_set_rule )
=> ( ( comple2146307154184993742t_rule @ ( set_or3245591702714615573t_rule @ X ) )
= top_top_set_rule ) ) ).
% Sup_greaterThanAtLeast
thf(fact_783_greaterThan__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_or575021546402375026an_rat @ K ) )
= ( ord_less_rat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_784_greaterThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_or1210151606488870762an_nat @ K ) )
= ( ord_less_nat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_785_greaterThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_or1210151606488870762an_nat @ X ) @ ( set_or1210151606488870762an_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% greaterThan_subset_iff
thf(fact_786_bijection_Obij__inv,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( bij_betw_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ top_top_set_rule @ top_top_set_rule ) ) ).
% bijection.bij_inv
thf(fact_787_bijection_Obij__inv,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( bij_betw_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat @ top_top_set_nat ) ) ).
% bijection.bij_inv
thf(fact_788_bijection_Obij__inv,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( bij_betw_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ top_top_set_rat @ top_top_set_rat ) ) ).
% bijection.bij_inv
thf(fact_789_bijection_Obij__inv,axiom,
! [F: set_rat > set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( bij_be8683898457559657236et_rat @ ( hilber3461591988630719082et_rat @ top_top_set_set_rat @ F ) @ top_top_set_set_rat @ top_top_set_set_rat ) ) ).
% bijection.bij_inv
thf(fact_790_bijection_Obij__inv,axiom,
! [F: set_nat > set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( bij_be3438014552859920132et_nat @ ( hilber7439080120785757786et_nat @ top_top_set_set_nat @ F ) @ top_top_set_set_nat @ top_top_set_set_nat ) ) ).
% bijection.bij_inv
thf(fact_791_bijection_Obij__inv,axiom,
! [F: set_rule > set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( bij_be2582266366990741742t_rule @ ( hilber824017158074262596t_rule @ top_top_set_set_rule @ F ) @ top_top_set_set_rule @ top_top_set_set_rule ) ) ).
% bijection.bij_inv
thf(fact_792_bij__betw__Pow,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( bij_betw_nat_rule @ F @ A @ B2 )
=> ( bij_be6872371095225530105t_rule @ ( image_nat_rule @ F ) @ ( pow_nat @ A ) @ ( pow_rule @ B2 ) ) ) ).
% bij_betw_Pow
thf(fact_793_bij__betw__Pow,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( bij_betw_nat_rat @ F @ A @ B2 )
=> ( bij_be9153158031321299212et_rat @ ( image_nat_rat @ F ) @ ( pow_nat @ A ) @ ( pow_rat @ B2 ) ) ) ).
% bij_betw_Pow
thf(fact_794_mono__onI,axiom,
! [A: set_rat,F: rat > nat] :
( ! [R: rat,S5: rat] :
( ( member_rat @ R @ A )
=> ( ( member_rat @ S5 @ A )
=> ( ( ord_less_eq_rat @ R @ S5 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ ord_less_eq_rat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_795_mono__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [R: nat,S5: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S5 @ A )
=> ( ( ord_less_eq_nat @ R @ S5 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_796_mono__onD,axiom,
! [A: set_rat,F: rat > nat,R2: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_eq_rat @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_797_mono__onD,axiom,
! [A: set_nat,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_eq_nat @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_798_ord_Omono__on__def,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > nat] :
( ( monotone_on_rat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: rat,S: rat] :
( ( ( member_rat @ R3 @ A )
& ( member_rat @ S @ A )
& ( Less_eq @ R3 @ S ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_799_ord_Omono__on__def,axiom,
! [A: set_rule,Less_eq: rule > rule > $o,F: rule > nat] :
( ( monotone_on_rule_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: rule,S: rule] :
( ( ( member_rule @ R3 @ A )
& ( member_rule @ S @ A )
& ( Less_eq @ R3 @ S ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_800_ord_Omono__on__def,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S @ A )
& ( Less_eq @ R3 @ S ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_801_ord_Omono__onI,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > nat] :
( ! [R: rat,S5: rat] :
( ( member_rat @ R @ A )
=> ( ( member_rat @ S5 @ A )
=> ( ( Less_eq @ R @ S5 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_802_ord_Omono__onI,axiom,
! [A: set_rule,Less_eq: rule > rule > $o,F: rule > nat] :
( ! [R: rule,S5: rule] :
( ( member_rule @ R @ A )
=> ( ( member_rule @ S5 @ A )
=> ( ( Less_eq @ R @ S5 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_rule_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_803_ord_Omono__onI,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R: nat,S5: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S5 @ A )
=> ( ( Less_eq @ R @ S5 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_804_ord_Omono__onD,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > nat,R2: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_805_ord_Omono__onD,axiom,
! [A: set_rule,Less_eq: rule > rule > $o,F: rule > nat,R2: rule,S2: rule] :
( ( monotone_on_rule_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_rule @ R2 @ A )
=> ( ( member_rule @ S2 @ A )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_806_ord_Omono__onD,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_807_strict__mono__on__eqD,axiom,
! [A: set_rat,F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_rat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_808_strict__mono__on__eqD,axiom,
! [A: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_809_strict__mono__onI,axiom,
! [A: set_rat,F: rat > nat] :
( ! [R: rat,S5: rat] :
( ( member_rat @ R @ A )
=> ( ( member_rat @ S5 @ A )
=> ( ( ord_less_rat @ R @ S5 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ ord_less_rat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_810_strict__mono__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [R: nat,S5: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S5 @ A )
=> ( ( ord_less_nat @ R @ S5 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_811_strict__mono__onD,axiom,
! [A: set_rat,F: rat > nat,R2: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_rat @ ord_less_nat @ F )
=> ( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_rat @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_812_strict__mono__onD,axiom,
! [A: set_nat,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_nat @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_813_ord_Ostrict__mono__on__def,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > nat] :
( ( monotone_on_rat_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: rat,S: rat] :
( ( ( member_rat @ R3 @ A )
& ( member_rat @ S @ A )
& ( Less @ R3 @ S ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_814_ord_Ostrict__mono__on__def,axiom,
! [A: set_rule,Less: rule > rule > $o,F: rule > nat] :
( ( monotone_on_rule_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: rule,S: rule] :
( ( ( member_rule @ R3 @ A )
& ( member_rule @ S @ A )
& ( Less @ R3 @ S ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_815_ord_Ostrict__mono__on__def,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: nat,S: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S @ A )
& ( Less @ R3 @ S ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_816_ord_Ostrict__mono__onI,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > nat] :
( ! [R: rat,S5: rat] :
( ( member_rat @ R @ A )
=> ( ( member_rat @ S5 @ A )
=> ( ( Less @ R @ S5 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_817_ord_Ostrict__mono__onI,axiom,
! [A: set_rule,Less: rule > rule > $o,F: rule > nat] :
( ! [R: rule,S5: rule] :
( ( member_rule @ R @ A )
=> ( ( member_rule @ S5 @ A )
=> ( ( Less @ R @ S5 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_rule_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_818_ord_Ostrict__mono__onI,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R: nat,S5: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S5 @ A )
=> ( ( Less @ R @ S5 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_819_ord_Ostrict__mono__onD,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > nat,R2: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_820_ord_Ostrict__mono__onD,axiom,
! [A: set_rule,Less: rule > rule > $o,F: rule > nat,R2: rule,S2: rule] :
( ( monotone_on_rule_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_rule @ R2 @ A )
=> ( ( member_rule @ S2 @ A )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_821_ord_Ostrict__mono__onD,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_822_bij__betw__iff__bijections,axiom,
( bij_betw_rat_rat
= ( ^ [F3: rat > rat,A5: set_rat,B5: set_rat] :
? [G2: rat > rat] :
( ! [X3: rat] :
( ( member_rat @ X3 @ A5 )
=> ( ( member_rat @ ( F3 @ X3 ) @ B5 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rat] :
( ( member_rat @ X3 @ B5 )
=> ( ( member_rat @ ( G2 @ X3 ) @ A5 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_823_bij__betw__iff__bijections,axiom,
( bij_betw_nat_rat
= ( ^ [F3: nat > rat,A5: set_nat,B5: set_rat] :
? [G2: rat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( member_rat @ ( F3 @ X3 ) @ B5 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rat] :
( ( member_rat @ X3 @ B5 )
=> ( ( member_nat @ ( G2 @ X3 ) @ A5 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_824_bij__betw__iff__bijections,axiom,
( bij_betw_rule_rat
= ( ^ [F3: rule > rat,A5: set_rule,B5: set_rat] :
? [G2: rat > rule] :
( ! [X3: rule] :
( ( member_rule @ X3 @ A5 )
=> ( ( member_rat @ ( F3 @ X3 ) @ B5 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rat] :
( ( member_rat @ X3 @ B5 )
=> ( ( member_rule @ ( G2 @ X3 ) @ A5 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_825_bij__betw__iff__bijections,axiom,
( bij_betw_rat_nat
= ( ^ [F3: rat > nat,A5: set_rat,B5: set_nat] :
? [G2: nat > rat] :
( ! [X3: rat] :
( ( member_rat @ X3 @ A5 )
=> ( ( member_nat @ ( F3 @ X3 ) @ B5 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( member_rat @ ( G2 @ X3 ) @ A5 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_826_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F3: nat > nat,A5: set_nat,B5: set_nat] :
? [G2: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ ( F3 @ X3 ) @ B5 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( member_nat @ ( G2 @ X3 ) @ A5 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_827_bij__betw__iff__bijections,axiom,
( bij_betw_rule_nat
= ( ^ [F3: rule > nat,A5: set_rule,B5: set_nat] :
? [G2: nat > rule] :
( ! [X3: rule] :
( ( member_rule @ X3 @ A5 )
=> ( ( member_nat @ ( F3 @ X3 ) @ B5 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( member_rule @ ( G2 @ X3 ) @ A5 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_828_bij__betw__iff__bijections,axiom,
( bij_betw_rat_rule
= ( ^ [F3: rat > rule,A5: set_rat,B5: set_rule] :
? [G2: rule > rat] :
( ! [X3: rat] :
( ( member_rat @ X3 @ A5 )
=> ( ( member_rule @ ( F3 @ X3 ) @ B5 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rule] :
( ( member_rule @ X3 @ B5 )
=> ( ( member_rat @ ( G2 @ X3 ) @ A5 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_829_bij__betw__iff__bijections,axiom,
( bij_betw_nat_rule
= ( ^ [F3: nat > rule,A5: set_nat,B5: set_rule] :
? [G2: rule > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( member_rule @ ( F3 @ X3 ) @ B5 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rule] :
( ( member_rule @ X3 @ B5 )
=> ( ( member_nat @ ( G2 @ X3 ) @ A5 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_830_bij__betw__iff__bijections,axiom,
( bij_betw_rule_rule
= ( ^ [F3: rule > rule,A5: set_rule,B5: set_rule] :
? [G2: rule > rule] :
( ! [X3: rule] :
( ( member_rule @ X3 @ A5 )
=> ( ( member_rule @ ( F3 @ X3 ) @ B5 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rule] :
( ( member_rule @ X3 @ B5 )
=> ( ( member_rule @ ( G2 @ X3 ) @ A5 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_831_monotone__on__def,axiom,
( monotone_on_nat_nat
= ( ^ [A5: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F3: nat > nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A5 )
=> ( ( Orda @ X3 @ Y2 )
=> ( Ordb @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_832_bij__betw__apply,axiom,
! [F: rat > rat,A: set_rat,B2: set_rat,A2: rat] :
( ( bij_betw_rat_rat @ F @ A @ B2 )
=> ( ( member_rat @ A2 @ A )
=> ( member_rat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_833_bij__betw__apply,axiom,
! [F: rat > nat,A: set_rat,B2: set_nat,A2: rat] :
( ( bij_betw_rat_nat @ F @ A @ B2 )
=> ( ( member_rat @ A2 @ A )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_834_bij__betw__apply,axiom,
! [F: rat > rule,A: set_rat,B2: set_rule,A2: rat] :
( ( bij_betw_rat_rule @ F @ A @ B2 )
=> ( ( member_rat @ A2 @ A )
=> ( member_rule @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_835_bij__betw__apply,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat,A2: nat] :
( ( bij_betw_nat_rat @ F @ A @ B2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_rat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_836_bij__betw__apply,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat,A2: nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_837_bij__betw__apply,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule,A2: nat] :
( ( bij_betw_nat_rule @ F @ A @ B2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_rule @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_838_bij__betw__apply,axiom,
! [F: rule > rat,A: set_rule,B2: set_rat,A2: rule] :
( ( bij_betw_rule_rat @ F @ A @ B2 )
=> ( ( member_rule @ A2 @ A )
=> ( member_rat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_839_bij__betw__apply,axiom,
! [F: rule > nat,A: set_rule,B2: set_nat,A2: rule] :
( ( bij_betw_rule_nat @ F @ A @ B2 )
=> ( ( member_rule @ A2 @ A )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_840_bij__betw__apply,axiom,
! [F: rule > rule,A: set_rule,B2: set_rule,A2: rule] :
( ( bij_betw_rule_rule @ F @ A @ B2 )
=> ( ( member_rule @ A2 @ A )
=> ( member_rule @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_841_monotone__onI,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat] :
( ! [X2: nat,Y5: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y5 @ A )
=> ( ( Orda2 @ X2 @ Y5 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Orda2 @ Ordb2 @ F ) ) ).
% monotone_onI
thf(fact_842_monotone__onD,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ Orda2 @ Ordb2 @ F )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( Orda2 @ X @ Y )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_843_monotone__on__subset,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat,B2: set_nat] :
( ( monotone_on_nat_nat @ A @ Orda2 @ Ordb2 @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( monotone_on_nat_nat @ B2 @ Orda2 @ Ordb2 @ F ) ) ) ).
% monotone_on_subset
thf(fact_844_bij__betw__imp__surj__on,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( bij_betw_nat_rule @ F @ A @ B2 )
=> ( ( image_nat_rule @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_845_bij__betw__imp__surj__on,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( bij_betw_nat_rat @ F @ A @ B2 )
=> ( ( image_nat_rat @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_846_monotoneD,axiom,
! [Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ Orda2 @ Ordb2 @ F )
=> ( ( Orda2 @ X @ Y )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_847_monotoneI,axiom,
! [Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat] :
( ! [X2: nat,Y5: nat] :
( ( Orda2 @ X2 @ Y5 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ Orda2 @ Ordb2 @ F ) ) ).
% monotoneI
thf(fact_848_bij__iff,axiom,
! [F: rule > rule] :
( ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule )
= ( ! [X3: rule] :
? [Y2: rule] :
( ( ( F @ Y2 )
= X3 )
& ! [Z5: rule] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_849_bij__iff,axiom,
! [F: rule > nat] :
( ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y2: rule] :
( ( ( F @ Y2 )
= X3 )
& ! [Z5: rule] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_850_bij__iff,axiom,
! [F: rule > rat] :
( ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat )
= ( ! [X3: rat] :
? [Y2: rule] :
( ( ( F @ Y2 )
= X3 )
& ! [Z5: rule] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_851_bij__iff,axiom,
! [F: nat > rule] :
( ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule )
= ( ! [X3: rule] :
? [Y2: nat] :
( ( ( F @ Y2 )
= X3 )
& ! [Z5: nat] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_852_bij__iff,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y2: nat] :
( ( ( F @ Y2 )
= X3 )
& ! [Z5: nat] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_853_bij__iff,axiom,
! [F: nat > rat] :
( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
= ( ! [X3: rat] :
? [Y2: nat] :
( ( ( F @ Y2 )
= X3 )
& ! [Z5: nat] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_854_bij__iff,axiom,
! [F: rat > rule] :
( ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule )
= ( ! [X3: rule] :
? [Y2: rat] :
( ( ( F @ Y2 )
= X3 )
& ! [Z5: rat] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_855_bij__iff,axiom,
! [F: rat > nat] :
( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y2: rat] :
( ( ( F @ Y2 )
= X3 )
& ! [Z5: rat] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_856_bij__iff,axiom,
! [F: rat > rat] :
( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
= ( ! [X3: rat] :
? [Y2: rat] :
( ( ( F @ Y2 )
= X3 )
& ! [Z5: rat] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_857_bij__iff,axiom,
! [F: rule > set_rat] :
( ( bij_be5592134826634264011et_rat @ F @ top_top_set_rule @ top_top_set_set_rat )
= ( ! [X3: set_rat] :
? [Y2: rule] :
( ( ( F @ Y2 )
= X3 )
& ! [Z5: rule] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_858_bij__pointE,axiom,
! [F: rule > rule,Y: rule] :
( ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule )
=> ~ ! [X2: rule] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X8: rule] :
( ( Y
= ( F @ X8 ) )
=> ( X8 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_859_bij__pointE,axiom,
! [F: rule > nat,Y: nat] :
( ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat )
=> ~ ! [X2: rule] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X8: rule] :
( ( Y
= ( F @ X8 ) )
=> ( X8 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_860_bij__pointE,axiom,
! [F: rule > rat,Y: rat] :
( ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat )
=> ~ ! [X2: rule] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X8: rule] :
( ( Y
= ( F @ X8 ) )
=> ( X8 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_861_bij__pointE,axiom,
! [F: nat > rule,Y: rule] :
( ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X8: nat] :
( ( Y
= ( F @ X8 ) )
=> ( X8 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_862_bij__pointE,axiom,
! [F: nat > nat,Y: nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X8: nat] :
( ( Y
= ( F @ X8 ) )
=> ( X8 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_863_bij__pointE,axiom,
! [F: nat > rat,Y: rat] :
( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X8: nat] :
( ( Y
= ( F @ X8 ) )
=> ( X8 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_864_bij__pointE,axiom,
! [F: rat > rule,Y: rule] :
( ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule )
=> ~ ! [X2: rat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X8: rat] :
( ( Y
= ( F @ X8 ) )
=> ( X8 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_865_bij__pointE,axiom,
! [F: rat > nat,Y: nat] :
( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
=> ~ ! [X2: rat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X8: rat] :
( ( Y
= ( F @ X8 ) )
=> ( X8 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_866_bij__pointE,axiom,
! [F: rat > rat,Y: rat] :
( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
=> ~ ! [X2: rat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X8: rat] :
( ( Y
= ( F @ X8 ) )
=> ( X8 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_867_bij__pointE,axiom,
! [F: rule > set_rat,Y: set_rat] :
( ( bij_be5592134826634264011et_rat @ F @ top_top_set_rule @ top_top_set_set_rat )
=> ~ ! [X2: rule] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X8: rule] :
( ( Y
= ( F @ X8 ) )
=> ( X8 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_868_involuntory__imp__bij,axiom,
! [F: rule > rule] :
( ! [X2: rule] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule ) ) ).
% involuntory_imp_bij
thf(fact_869_involuntory__imp__bij,axiom,
! [F: nat > nat] :
( ! [X2: nat] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ).
% involuntory_imp_bij
thf(fact_870_involuntory__imp__bij,axiom,
! [F: rat > rat] :
( ! [X2: rat] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat ) ) ).
% involuntory_imp_bij
thf(fact_871_involuntory__imp__bij,axiom,
! [F: set_rat > set_rat] :
( ! [X2: set_rat] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_be8683898457559657236et_rat @ F @ top_top_set_set_rat @ top_top_set_set_rat ) ) ).
% involuntory_imp_bij
thf(fact_872_involuntory__imp__bij,axiom,
! [F: set_nat > set_nat] :
( ! [X2: set_nat] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_be3438014552859920132et_nat @ F @ top_top_set_set_nat @ top_top_set_set_nat ) ) ).
% involuntory_imp_bij
thf(fact_873_involuntory__imp__bij,axiom,
! [F: set_rule > set_rule] :
( ! [X2: set_rule] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_be2582266366990741742t_rule @ F @ top_top_set_set_rule @ top_top_set_set_rule ) ) ).
% involuntory_imp_bij
thf(fact_874_bijection_Obij,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule ) ) ).
% bijection.bij
thf(fact_875_bijection_Obij,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ).
% bijection.bij
thf(fact_876_bijection_Obij,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat ) ) ).
% bijection.bij
thf(fact_877_bijection_Obij,axiom,
! [F: set_rat > set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( bij_be8683898457559657236et_rat @ F @ top_top_set_set_rat @ top_top_set_set_rat ) ) ).
% bijection.bij
thf(fact_878_bijection_Obij,axiom,
! [F: set_nat > set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( bij_be3438014552859920132et_nat @ F @ top_top_set_set_nat @ top_top_set_set_nat ) ) ).
% bijection.bij
thf(fact_879_bijection_Obij,axiom,
! [F: set_rule > set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( bij_be2582266366990741742t_rule @ F @ top_top_set_set_rule @ top_top_set_set_rule ) ) ).
% bijection.bij
thf(fact_880_bijection_Ointro,axiom,
! [F: rule > rule] :
( ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule )
=> ( hilber6733072011887318294n_rule @ F ) ) ).
% bijection.intro
thf(fact_881_bijection_Ointro,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( hilber5277034221543178913on_nat @ F ) ) ).
% bijection.intro
thf(fact_882_bijection_Ointro,axiom,
! [F: rat > rat] :
( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
=> ( hilber4641904161456683177on_rat @ F ) ) ).
% bijection.intro
thf(fact_883_bijection_Ointro,axiom,
! [F: set_rat > set_rat] :
( ( bij_be8683898457559657236et_rat @ F @ top_top_set_set_rat @ top_top_set_set_rat )
=> ( hilber3240988290065685599et_rat @ F ) ) ).
% bijection.intro
thf(fact_884_bijection_Ointro,axiom,
! [F: set_nat > set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ top_top_set_set_nat @ top_top_set_set_nat )
=> ( hilber6749216848459082327et_nat @ F ) ) ).
% bijection.intro
thf(fact_885_bijection_Ointro,axiom,
! [F: set_rule > set_rule] :
( ( bij_be2582266366990741742t_rule @ F @ top_top_set_set_rule @ top_top_set_set_rule )
=> ( hilber5661084989754287948t_rule @ F ) ) ).
% bijection.intro
thf(fact_886_bijection__def,axiom,
( hilber6733072011887318294n_rule
= ( ^ [F3: rule > rule] : ( bij_betw_rule_rule @ F3 @ top_top_set_rule @ top_top_set_rule ) ) ) ).
% bijection_def
thf(fact_887_bijection__def,axiom,
( hilber5277034221543178913on_nat
= ( ^ [F3: nat > nat] : ( bij_betw_nat_nat @ F3 @ top_top_set_nat @ top_top_set_nat ) ) ) ).
% bijection_def
thf(fact_888_bijection__def,axiom,
( hilber4641904161456683177on_rat
= ( ^ [F3: rat > rat] : ( bij_betw_rat_rat @ F3 @ top_top_set_rat @ top_top_set_rat ) ) ) ).
% bijection_def
thf(fact_889_bijection__def,axiom,
( hilber3240988290065685599et_rat
= ( ^ [F3: set_rat > set_rat] : ( bij_be8683898457559657236et_rat @ F3 @ top_top_set_set_rat @ top_top_set_set_rat ) ) ) ).
% bijection_def
thf(fact_890_bijection__def,axiom,
( hilber6749216848459082327et_nat
= ( ^ [F3: set_nat > set_nat] : ( bij_be3438014552859920132et_nat @ F3 @ top_top_set_set_nat @ top_top_set_set_nat ) ) ) ).
% bijection_def
thf(fact_891_bijection__def,axiom,
( hilber5661084989754287948t_rule
= ( ^ [F3: set_rule > set_rule] : ( bij_be2582266366990741742t_rule @ F3 @ top_top_set_set_rule @ top_top_set_set_rule ) ) ) ).
% bijection_def
thf(fact_892_mono__inv,axiom,
! [F: rat > rat] :
( ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_rat @ F )
=> ( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
=> ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) ) ) ) ).
% mono_inv
thf(fact_893_mono__inv,axiom,
! [F: rat > nat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
=> ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) ) ) ) ).
% mono_inv
thf(fact_894_mono__inv,axiom,
! [F: nat > rat] :
( ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_rat @ F )
=> ( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) ) ) ) ).
% mono_inv
thf(fact_895_mono__inv,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) ) ) ) ).
% mono_inv
thf(fact_896_mono__on__greaterD,axiom,
! [A: set_rat,G: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_eq_rat @ ord_less_eq_nat @ G )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_rat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_897_mono__on__greaterD,axiom,
! [A: set_nat,G: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ G )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_898_strict__mono__on__leD,axiom,
! [A: set_rat,F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_rat @ ord_less_nat @ F )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_899_strict__mono__on__leD,axiom,
! [A: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_900_strict__mono__on__imp__mono__on,axiom,
! [A: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_901_mono__imp__mono__on,axiom,
! [F: rat > nat,A: set_rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( monotone_on_rat_nat @ A @ ord_less_eq_rat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_902_mono__imp__mono__on,axiom,
! [F: set_rat > nat,A: set_set_rat] :
( ( monoto7746382625157022407at_nat @ top_top_set_set_rat @ ord_less_eq_set_rat @ ord_less_eq_nat @ F )
=> ( monoto7746382625157022407at_nat @ A @ ord_less_eq_set_rat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_903_mono__imp__mono__on,axiom,
! [F: set_nat > nat,A: set_set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( monoto2923694778811248831at_nat @ A @ ord_less_eq_set_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_904_mono__imp__mono__on,axiom,
! [F: set_rule > nat,A: set_set_rule] :
( ( monoto593345095951663796le_nat @ top_top_set_set_rule @ ord_less_eq_set_rule @ ord_less_eq_nat @ F )
=> ( monoto593345095951663796le_nat @ A @ ord_less_eq_set_rule @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_905_mono__imp__mono__on,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_906_monoI,axiom,
! [F: rat > nat] :
( ! [X2: rat,Y5: rat] :
( ( ord_less_eq_rat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_907_monoI,axiom,
! [F: set_rat > nat] :
( ! [X2: set_rat,Y5: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( monoto7746382625157022407at_nat @ top_top_set_set_rat @ ord_less_eq_set_rat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_908_monoI,axiom,
! [F: set_nat > nat] :
( ! [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_909_monoI,axiom,
! [F: set_rule > nat] :
( ! [X2: set_rule,Y5: set_rule] :
( ( ord_less_eq_set_rule @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( monoto593345095951663796le_nat @ top_top_set_set_rule @ ord_less_eq_set_rule @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_910_monoI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_911_monoE,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_912_monoE,axiom,
! [F: set_rat > nat,X: set_rat,Y: set_rat] :
( ( monoto7746382625157022407at_nat @ top_top_set_set_rat @ ord_less_eq_set_rat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_913_monoE,axiom,
! [F: set_nat > nat,X: set_nat,Y: set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_914_monoE,axiom,
! [F: set_rule > nat,X: set_rule,Y: set_rule] :
( ( monoto593345095951663796le_nat @ top_top_set_set_rule @ ord_less_eq_set_rule @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_rule @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_915_monoE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_916_monoD,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_917_monoD,axiom,
! [F: set_rat > nat,X: set_rat,Y: set_rat] :
( ( monoto7746382625157022407at_nat @ top_top_set_set_rat @ ord_less_eq_set_rat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_918_monoD,axiom,
! [F: set_nat > nat,X: set_nat,Y: set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_919_monoD,axiom,
! [F: set_rule > nat,X: set_rule,Y: set_rule] :
( ( monoto593345095951663796le_nat @ top_top_set_set_rule @ ord_less_eq_set_rule @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_rule @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_920_monoD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_921_strict__mono__less,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_rat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_922_strict__mono__less,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_923_strict__mono__eq,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_924_strict__mono__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_925_strict__monoI,axiom,
! [F: rat > nat] :
( ! [X2: rat,Y5: rat] :
( ( ord_less_rat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_926_strict__monoI,axiom,
! [F: set_rat > nat] :
( ! [X2: set_rat,Y5: set_rat] :
( ( ord_less_set_rat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( monoto7746382625157022407at_nat @ top_top_set_set_rat @ ord_less_set_rat @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_927_strict__monoI,axiom,
! [F: set_nat > nat] :
( ! [X2: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_set_nat @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_928_strict__monoI,axiom,
! [F: set_rule > nat] :
( ! [X2: set_rule,Y5: set_rule] :
( ( ord_less_set_rule @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( monoto593345095951663796le_nat @ top_top_set_set_rule @ ord_less_set_rule @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_929_strict__monoI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_930_strict__monoD,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( ( ord_less_rat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_931_strict__monoD,axiom,
! [F: set_rat > nat,X: set_rat,Y: set_rat] :
( ( monoto7746382625157022407at_nat @ top_top_set_set_rat @ ord_less_set_rat @ ord_less_nat @ F )
=> ( ( ord_less_set_rat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_932_strict__monoD,axiom,
! [F: set_nat > nat,X: set_nat,Y: set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_set_nat @ ord_less_nat @ F )
=> ( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_933_strict__monoD,axiom,
! [F: set_rule > nat,X: set_rule,Y: set_rule] :
( ( monoto593345095951663796le_nat @ top_top_set_set_rule @ ord_less_set_rule @ ord_less_nat @ F )
=> ( ( ord_less_set_rule @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_934_strict__monoD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_935_ord_Omono__on__subset,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat,B2: set_nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( monotone_on_nat_nat @ B2 @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_936_mono__on__subset,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( monotone_on_nat_nat @ B2 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_937_bij__diff,axiom,
! [A2: rat] : ( bij_betw_rat_rat @ ( minus_minus_rat @ A2 ) @ top_top_set_rat @ top_top_set_rat ) ).
% bij_diff
thf(fact_938_bij__betw__imp__surj,axiom,
! [F: rule > rule,A: set_rule] :
( ( bij_betw_rule_rule @ F @ A @ top_top_set_rule )
=> ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule ) ) ).
% bij_betw_imp_surj
thf(fact_939_bij__betw__imp__surj,axiom,
! [F: nat > rule,A: set_nat] :
( ( bij_betw_nat_rule @ F @ A @ top_top_set_rule )
=> ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule ) ) ).
% bij_betw_imp_surj
thf(fact_940_bij__betw__imp__surj,axiom,
! [F: rat > rule,A: set_rat] :
( ( bij_betw_rat_rule @ F @ A @ top_top_set_rule )
=> ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule ) ) ).
% bij_betw_imp_surj
thf(fact_941_bij__betw__imp__surj,axiom,
! [F: rule > nat,A: set_rule] :
( ( bij_betw_rule_nat @ F @ A @ top_top_set_nat )
=> ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat ) ) ).
% bij_betw_imp_surj
thf(fact_942_bij__betw__imp__surj,axiom,
! [F: nat > nat,A: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bij_betw_imp_surj
thf(fact_943_bij__betw__imp__surj,axiom,
! [F: rat > nat,A: set_rat] :
( ( bij_betw_rat_nat @ F @ A @ top_top_set_nat )
=> ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat ) ) ).
% bij_betw_imp_surj
thf(fact_944_bij__betw__imp__surj,axiom,
! [F: rule > rat,A: set_rule] :
( ( bij_betw_rule_rat @ F @ A @ top_top_set_rat )
=> ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat ) ) ).
% bij_betw_imp_surj
thf(fact_945_bij__betw__imp__surj,axiom,
! [F: nat > rat,A: set_nat] :
( ( bij_betw_nat_rat @ F @ A @ top_top_set_rat )
=> ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat ) ) ).
% bij_betw_imp_surj
thf(fact_946_bij__betw__imp__surj,axiom,
! [F: rat > rat,A: set_rat] :
( ( bij_betw_rat_rat @ F @ A @ top_top_set_rat )
=> ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat ) ) ).
% bij_betw_imp_surj
thf(fact_947_bij__betw__imp__surj,axiom,
! [F: set_rat > rule,A: set_set_rat] :
( ( bij_be2386713868094972619t_rule @ F @ A @ top_top_set_rule )
=> ( ( image_set_rat_rule @ F @ top_top_set_set_rat )
= top_top_set_rule ) ) ).
% bij_betw_imp_surj
thf(fact_948_bij__is__surj,axiom,
! [F: rule > rule] :
( ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule )
=> ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule ) ) ).
% bij_is_surj
thf(fact_949_bij__is__surj,axiom,
! [F: rule > nat] :
( ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat )
=> ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat ) ) ).
% bij_is_surj
thf(fact_950_bij__is__surj,axiom,
! [F: rule > rat] :
( ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat )
=> ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat ) ) ).
% bij_is_surj
thf(fact_951_bij__is__surj,axiom,
! [F: nat > rule] :
( ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule )
=> ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule ) ) ).
% bij_is_surj
thf(fact_952_bij__is__surj,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bij_is_surj
thf(fact_953_bij__is__surj,axiom,
! [F: nat > rat] :
( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
=> ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat ) ) ).
% bij_is_surj
thf(fact_954_bij__is__surj,axiom,
! [F: rat > rule] :
( ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule )
=> ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule ) ) ).
% bij_is_surj
thf(fact_955_bij__is__surj,axiom,
! [F: rat > nat] :
( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
=> ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat ) ) ).
% bij_is_surj
thf(fact_956_bij__is__surj,axiom,
! [F: rat > rat] :
( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
=> ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat ) ) ).
% bij_is_surj
thf(fact_957_bij__is__surj,axiom,
! [F: rule > set_rat] :
( ( bij_be5592134826634264011et_rat @ F @ top_top_set_rule @ top_top_set_set_rat )
=> ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat ) ) ).
% bij_is_surj
thf(fact_958_Fun_Obij__uminus,axiom,
bij_betw_rat_rat @ uminus_uminus_rat @ top_top_set_rat @ top_top_set_rat ).
% Fun.bij_uminus
thf(fact_959_bij__betw__subset,axiom,
! [F: nat > rule,A: set_nat,A7: set_rule,B2: set_nat,B7: set_rule] :
( ( bij_betw_nat_rule @ F @ A @ A7 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ( image_nat_rule @ F @ B2 )
= B7 )
=> ( bij_betw_nat_rule @ F @ B2 @ B7 ) ) ) ) ).
% bij_betw_subset
thf(fact_960_bij__betw__subset,axiom,
! [F: nat > rat,A: set_nat,A7: set_rat,B2: set_nat,B7: set_rat] :
( ( bij_betw_nat_rat @ F @ A @ A7 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ( image_nat_rat @ F @ B2 )
= B7 )
=> ( bij_betw_nat_rat @ F @ B2 @ B7 ) ) ) ) ).
% bij_betw_subset
thf(fact_961_bij__betw__byWitness,axiom,
! [A: set_rule,F4: nat > rule,F: rule > nat,A7: set_nat] :
( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( ( F4 @ ( F @ X2 ) )
= X2 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A7 )
=> ( ( F @ ( F4 @ X2 ) )
= X2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_rule_nat @ F @ A ) @ A7 )
=> ( ( ord_less_eq_set_rule @ ( image_nat_rule @ F4 @ A7 ) @ A )
=> ( bij_betw_rule_nat @ F @ A @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_962_bij__betw__byWitness,axiom,
! [A: set_rat,F4: nat > rat,F: rat > nat,A7: set_nat] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ( F4 @ ( F @ X2 ) )
= X2 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A7 )
=> ( ( F @ ( F4 @ X2 ) )
= X2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_rat_nat @ F @ A ) @ A7 )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F4 @ A7 ) @ A )
=> ( bij_betw_rat_nat @ F @ A @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_963_bij__betw__byWitness,axiom,
! [A: set_nat,F4: rule > nat,F: nat > rule,A7: set_rule] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( F4 @ ( F @ X2 ) )
= X2 ) )
=> ( ! [X2: rule] :
( ( member_rule @ X2 @ A7 )
=> ( ( F @ ( F4 @ X2 ) )
= X2 ) )
=> ( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ A7 )
=> ( ( ord_less_eq_set_nat @ ( image_rule_nat @ F4 @ A7 ) @ A )
=> ( bij_betw_nat_rule @ F @ A @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_964_bij__betw__byWitness,axiom,
! [A: set_nat,F4: rat > nat,F: nat > rat,A7: set_rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( F4 @ ( F @ X2 ) )
= X2 ) )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ A7 )
=> ( ( F @ ( F4 @ X2 ) )
= X2 ) )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ A7 )
=> ( ( ord_less_eq_set_nat @ ( image_rat_nat @ F4 @ A7 ) @ A )
=> ( bij_betw_nat_rat @ F @ A @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_965_strict__mono__on__imp__inj__on,axiom,
! [A: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% strict_mono_on_imp_inj_on
thf(fact_966_bij__is__inj,axiom,
! [F: rule > rule] :
( ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule )
=> ( inj_on_rule_rule @ F @ top_top_set_rule ) ) ).
% bij_is_inj
thf(fact_967_bij__is__inj,axiom,
! [F: rule > nat] :
( ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat )
=> ( inj_on_rule_nat @ F @ top_top_set_rule ) ) ).
% bij_is_inj
thf(fact_968_bij__is__inj,axiom,
! [F: rule > rat] :
( ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat )
=> ( inj_on_rule_rat @ F @ top_top_set_rule ) ) ).
% bij_is_inj
thf(fact_969_bij__is__inj,axiom,
! [F: nat > rule] :
( ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule )
=> ( inj_on_nat_rule @ F @ top_top_set_nat ) ) ).
% bij_is_inj
thf(fact_970_bij__is__inj,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% bij_is_inj
thf(fact_971_bij__is__inj,axiom,
! [F: nat > rat] :
( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
=> ( inj_on_nat_rat @ F @ top_top_set_nat ) ) ).
% bij_is_inj
thf(fact_972_bij__is__inj,axiom,
! [F: rat > rule] :
( ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule )
=> ( inj_on_rat_rule @ F @ top_top_set_rat ) ) ).
% bij_is_inj
thf(fact_973_bij__is__inj,axiom,
! [F: rat > nat] :
( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
=> ( inj_on_rat_nat @ F @ top_top_set_rat ) ) ).
% bij_is_inj
thf(fact_974_bij__is__inj,axiom,
! [F: rat > rat] :
( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
=> ( inj_on_rat_rat @ F @ top_top_set_rat ) ) ).
% bij_is_inj
thf(fact_975_bij__is__inj,axiom,
! [F: rule > set_rat] :
( ( bij_be5592134826634264011et_rat @ F @ top_top_set_rule @ top_top_set_set_rat )
=> ( inj_on_rule_set_rat @ F @ top_top_set_rule ) ) ).
% bij_is_inj
thf(fact_976_bij__betw__def,axiom,
( bij_betw_nat_rule
= ( ^ [F3: nat > rule,A5: set_nat,B5: set_rule] :
( ( inj_on_nat_rule @ F3 @ A5 )
& ( ( image_nat_rule @ F3 @ A5 )
= B5 ) ) ) ) ).
% bij_betw_def
thf(fact_977_bij__betw__def,axiom,
( bij_betw_nat_rat
= ( ^ [F3: nat > rat,A5: set_nat,B5: set_rat] :
( ( inj_on_nat_rat @ F3 @ A5 )
& ( ( image_nat_rat @ F3 @ A5 )
= B5 ) ) ) ) ).
% bij_betw_def
thf(fact_978_bij__betw__imageI,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( ( image_nat_rule @ F @ A )
= B2 )
=> ( bij_betw_nat_rule @ F @ A @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_979_bij__betw__imageI,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( ( image_nat_rat @ F @ A )
= B2 )
=> ( bij_betw_nat_rat @ F @ A @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_980_inj__on__imp__bij__betw,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( bij_betw_nat_rule @ F @ A @ ( image_nat_rule @ F @ A ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_981_inj__on__imp__bij__betw,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( bij_betw_nat_rat @ F @ A @ ( image_nat_rat @ F @ A ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_982_incseq__def,axiom,
! [X4: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X4 )
= ( ! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( X4 @ M ) @ ( X4 @ N ) ) ) ) ) ).
% incseq_def
thf(fact_983_incseqD,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% incseqD
thf(fact_984_inv__inv__eq,axiom,
! [F: rule > rule] :
( ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule )
=> ( ( hilber2978553400015838680e_rule @ top_top_set_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) )
= F ) ) ).
% inv_inv_eq
thf(fact_985_inv__inv__eq,axiom,
! [F: rule > nat] :
( ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat )
=> ( ( hilber8541579349336805475t_rule @ top_top_set_nat @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F ) )
= F ) ) ).
% inv_inv_eq
thf(fact_986_inv__inv__eq,axiom,
! [F: rule > rat] :
( ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat )
=> ( ( hilber5214430877627997803t_rule @ top_top_set_rat @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F ) )
= F ) ) ).
% inv_inv_eq
thf(fact_987_inv__inv__eq,axiom,
! [F: nat > rule] :
( ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule )
=> ( ( hilber2555471727301889379le_nat @ top_top_set_rule @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) )
= F ) ) ).
% inv_inv_eq
thf(fact_988_inv__inv__eq,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) )
= F ) ) ).
% inv_inv_eq
thf(fact_989_inv__inv__eq,axiom,
! [F: nat > rat] :
( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
=> ( ( hilber3317322552863949046at_nat @ top_top_set_rat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) )
= F ) ) ).
% inv_inv_eq
thf(fact_990_inv__inv__eq,axiom,
! [F: rat > rule] :
( ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule )
=> ( ( hilber1920341667215393643le_rat @ top_top_set_rule @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F ) )
= F ) ) ).
% inv_inv_eq
thf(fact_991_inv__inv__eq,axiom,
! [F: rat > nat] :
( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
=> ( ( hilber2998747136712319222at_rat @ top_top_set_nat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) )
= F ) ) ).
% inv_inv_eq
thf(fact_992_inv__inv__eq,axiom,
! [F: rat > rat] :
( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
=> ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) )
= F ) ) ).
% inv_inv_eq
thf(fact_993_inv__inv__eq,axiom,
! [F: rule > set_rat] :
( ( bij_be5592134826634264011et_rat @ F @ top_top_set_rule @ top_top_set_set_rat )
=> ( ( hilber3948050505403073569t_rule @ top_top_set_set_rat @ ( hilber7153471463942364961et_rat @ top_top_set_rule @ F ) )
= F ) ) ).
% inv_inv_eq
thf(fact_994_bij__inv__eq__iff,axiom,
! [P5: rule > rule,X: rule,Y: rule] :
( ( bij_betw_rule_rule @ P5 @ top_top_set_rule @ top_top_set_rule )
=> ( ( X
= ( hilber2978553400015838680e_rule @ top_top_set_rule @ P5 @ Y ) )
= ( ( P5 @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_995_bij__inv__eq__iff,axiom,
! [P5: rule > nat,X: rule,Y: nat] :
( ( bij_betw_rule_nat @ P5 @ top_top_set_rule @ top_top_set_nat )
=> ( ( X
= ( hilber2555471727301889379le_nat @ top_top_set_rule @ P5 @ Y ) )
= ( ( P5 @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_996_bij__inv__eq__iff,axiom,
! [P5: rule > rat,X: rule,Y: rat] :
( ( bij_betw_rule_rat @ P5 @ top_top_set_rule @ top_top_set_rat )
=> ( ( X
= ( hilber1920341667215393643le_rat @ top_top_set_rule @ P5 @ Y ) )
= ( ( P5 @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_997_bij__inv__eq__iff,axiom,
! [P5: nat > rule,X: nat,Y: rule] :
( ( bij_betw_nat_rule @ P5 @ top_top_set_nat @ top_top_set_rule )
=> ( ( X
= ( hilber8541579349336805475t_rule @ top_top_set_nat @ P5 @ Y ) )
= ( ( P5 @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_998_bij__inv__eq__iff,axiom,
! [P5: nat > nat,X: nat,Y: nat] :
( ( bij_betw_nat_nat @ P5 @ top_top_set_nat @ top_top_set_nat )
=> ( ( X
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ P5 @ Y ) )
= ( ( P5 @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_999_bij__inv__eq__iff,axiom,
! [P5: nat > rat,X: nat,Y: rat] :
( ( bij_betw_nat_rat @ P5 @ top_top_set_nat @ top_top_set_rat )
=> ( ( X
= ( hilber2998747136712319222at_rat @ top_top_set_nat @ P5 @ Y ) )
= ( ( P5 @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_1000_bij__inv__eq__iff,axiom,
! [P5: rat > rule,X: rat,Y: rule] :
( ( bij_betw_rat_rule @ P5 @ top_top_set_rat @ top_top_set_rule )
=> ( ( X
= ( hilber5214430877627997803t_rule @ top_top_set_rat @ P5 @ Y ) )
= ( ( P5 @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_1001_bij__inv__eq__iff,axiom,
! [P5: rat > nat,X: rat,Y: nat] :
( ( bij_betw_rat_nat @ P5 @ top_top_set_rat @ top_top_set_nat )
=> ( ( X
= ( hilber3317322552863949046at_nat @ top_top_set_rat @ P5 @ Y ) )
= ( ( P5 @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_1002_bij__inv__eq__iff,axiom,
! [P5: rat > rat,X: rat,Y: rat] :
( ( bij_betw_rat_rat @ P5 @ top_top_set_rat @ top_top_set_rat )
=> ( ( X
= ( hilber2682192492777453310at_rat @ top_top_set_rat @ P5 @ Y ) )
= ( ( P5 @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_1003_bij__inv__eq__iff,axiom,
! [P5: rule > set_rat,X: rule,Y: set_rat] :
( ( bij_be5592134826634264011et_rat @ P5 @ top_top_set_rule @ top_top_set_set_rat )
=> ( ( X
= ( hilber7153471463942364961et_rat @ top_top_set_rule @ P5 @ Y ) )
= ( ( P5 @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_1004_bij__imp__bij__inv,axiom,
! [F: rule > rule] :
( ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule )
=> ( bij_betw_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ top_top_set_rule @ top_top_set_rule ) ) ).
% bij_imp_bij_inv
thf(fact_1005_bij__imp__bij__inv,axiom,
! [F: rule > nat] :
( ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat )
=> ( bij_betw_nat_rule @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F ) @ top_top_set_nat @ top_top_set_rule ) ) ).
% bij_imp_bij_inv
thf(fact_1006_bij__imp__bij__inv,axiom,
! [F: rule > rat] :
( ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat )
=> ( bij_betw_rat_rule @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F ) @ top_top_set_rat @ top_top_set_rule ) ) ).
% bij_imp_bij_inv
thf(fact_1007_bij__imp__bij__inv,axiom,
! [F: nat > rule] :
( ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule )
=> ( bij_betw_rule_nat @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) @ top_top_set_rule @ top_top_set_nat ) ) ).
% bij_imp_bij_inv
thf(fact_1008_bij__imp__bij__inv,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( bij_betw_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat @ top_top_set_nat ) ) ).
% bij_imp_bij_inv
thf(fact_1009_bij__imp__bij__inv,axiom,
! [F: nat > rat] :
( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
=> ( bij_betw_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) @ top_top_set_rat @ top_top_set_nat ) ) ).
% bij_imp_bij_inv
thf(fact_1010_bij__imp__bij__inv,axiom,
! [F: rat > rule] :
( ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule )
=> ( bij_betw_rule_rat @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F ) @ top_top_set_rule @ top_top_set_rat ) ) ).
% bij_imp_bij_inv
thf(fact_1011_bij__imp__bij__inv,axiom,
! [F: rat > nat] :
( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
=> ( bij_betw_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) @ top_top_set_nat @ top_top_set_rat ) ) ).
% bij_imp_bij_inv
thf(fact_1012_bij__imp__bij__inv,axiom,
! [F: rat > rat] :
( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
=> ( bij_betw_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ top_top_set_rat @ top_top_set_rat ) ) ).
% bij_imp_bij_inv
thf(fact_1013_bij__imp__bij__inv,axiom,
! [F: rule > set_rat] :
( ( bij_be5592134826634264011et_rat @ F @ top_top_set_rule @ top_top_set_set_rat )
=> ( bij_be2386713868094972619t_rule @ ( hilber7153471463942364961et_rat @ top_top_set_rule @ F ) @ top_top_set_set_rat @ top_top_set_rule ) ) ).
% bij_imp_bij_inv
thf(fact_1014_strict__mono__less__eq,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_1015_strict__mono__less__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_1016_mono__strict__invE,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_rat @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_1017_mono__strict__invE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_1018_strict__mono__mono,axiom,
! [F: rat > nat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_1019_strict__mono__mono,axiom,
! [F: set_rat > nat] :
( ( monoto7746382625157022407at_nat @ top_top_set_set_rat @ ord_less_set_rat @ ord_less_nat @ F )
=> ( monoto7746382625157022407at_nat @ top_top_set_set_rat @ ord_less_eq_set_rat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_1020_strict__mono__mono,axiom,
! [F: set_nat > nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_set_nat @ ord_less_nat @ F )
=> ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_1021_strict__mono__mono,axiom,
! [F: set_rule > nat] :
( ( monoto593345095951663796le_nat @ top_top_set_set_rule @ ord_less_set_rule @ ord_less_nat @ F )
=> ( monoto593345095951663796le_nat @ top_top_set_set_rule @ ord_less_eq_set_rule @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_1022_strict__mono__mono,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_1023_mono__invE,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1024_mono__invE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1025_strict__mono__leD,axiom,
! [R2: rat > nat,M3: rat,N3: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_rat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( R2 @ M3 ) @ ( R2 @ N3 ) ) ) ) ).
% strict_mono_leD
thf(fact_1026_strict__mono__leD,axiom,
! [R2: set_rat > nat,M3: set_rat,N3: set_rat] :
( ( monoto7746382625157022407at_nat @ top_top_set_set_rat @ ord_less_set_rat @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_set_rat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( R2 @ M3 ) @ ( R2 @ N3 ) ) ) ) ).
% strict_mono_leD
thf(fact_1027_strict__mono__leD,axiom,
! [R2: set_nat > nat,M3: set_nat,N3: set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_set_nat @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_set_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( R2 @ M3 ) @ ( R2 @ N3 ) ) ) ) ).
% strict_mono_leD
thf(fact_1028_strict__mono__leD,axiom,
! [R2: set_rule > nat,M3: set_rule,N3: set_rule] :
( ( monoto593345095951663796le_nat @ top_top_set_set_rule @ ord_less_set_rule @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_set_rule @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( R2 @ M3 ) @ ( R2 @ N3 ) ) ) ) ).
% strict_mono_leD
thf(fact_1029_strict__mono__leD,axiom,
! [R2: nat > nat,M3: nat,N3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( R2 @ M3 ) @ ( R2 @ N3 ) ) ) ) ).
% strict_mono_leD
thf(fact_1030_strict__mono__inv,axiom,
! [F: rat > rat,G: rat > rat] :
( ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_rat @ ord_less_rat @ F )
=> ( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_rat @ ord_less_rat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1031_strict__mono__inv,axiom,
! [F: rat > nat,G: nat > rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_nat @ ord_less_rat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1032_strict__mono__inv,axiom,
! [F: nat > rat,G: rat > nat] :
( ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_nat @ ord_less_rat @ F )
=> ( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1033_strict__mono__inv,axiom,
! [F: nat > nat,G: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1034_strict__mono__imp__inj__on,axiom,
! [F: rat > nat,A: set_rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( inj_on_rat_nat @ F @ A ) ) ).
% strict_mono_imp_inj_on
thf(fact_1035_strict__mono__imp__inj__on,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% strict_mono_imp_inj_on
thf(fact_1036_bij__def,axiom,
! [F: rule > rule] :
( ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule )
= ( ( inj_on_rule_rule @ F @ top_top_set_rule )
& ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule ) ) ) ).
% bij_def
thf(fact_1037_bij__def,axiom,
! [F: rule > nat] :
( ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat )
= ( ( inj_on_rule_nat @ F @ top_top_set_rule )
& ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat ) ) ) ).
% bij_def
thf(fact_1038_bij__def,axiom,
! [F: rule > rat] :
( ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat )
= ( ( inj_on_rule_rat @ F @ top_top_set_rule )
& ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat ) ) ) ).
% bij_def
thf(fact_1039_bij__def,axiom,
! [F: nat > rule] :
( ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule )
= ( ( inj_on_nat_rule @ F @ top_top_set_nat )
& ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule ) ) ) ).
% bij_def
thf(fact_1040_bij__def,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
= ( ( inj_on_nat_nat @ F @ top_top_set_nat )
& ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% bij_def
thf(fact_1041_bij__def,axiom,
! [F: nat > rat] :
( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
= ( ( inj_on_nat_rat @ F @ top_top_set_nat )
& ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat ) ) ) ).
% bij_def
thf(fact_1042_bij__def,axiom,
! [F: rat > rule] :
( ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule )
= ( ( inj_on_rat_rule @ F @ top_top_set_rat )
& ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule ) ) ) ).
% bij_def
thf(fact_1043_bij__def,axiom,
! [F: rat > nat] :
( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
= ( ( inj_on_rat_nat @ F @ top_top_set_rat )
& ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat ) ) ) ).
% bij_def
thf(fact_1044_bij__def,axiom,
! [F: rat > rat] :
( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
= ( ( inj_on_rat_rat @ F @ top_top_set_rat )
& ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat ) ) ) ).
% bij_def
thf(fact_1045_bij__def,axiom,
! [F: rule > set_rat] :
( ( bij_be5592134826634264011et_rat @ F @ top_top_set_rule @ top_top_set_set_rat )
= ( ( inj_on_rule_set_rat @ F @ top_top_set_rule )
& ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat ) ) ) ).
% bij_def
thf(fact_1046_bijI,axiom,
! [F: rule > rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule ) ) ) ).
% bijI
thf(fact_1047_bijI,axiom,
! [F: rule > nat] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat ) ) ) ).
% bijI
thf(fact_1048_bijI,axiom,
! [F: rule > rat] :
( ( inj_on_rule_rat @ F @ top_top_set_rule )
=> ( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat ) ) ) ).
% bijI
thf(fact_1049_bijI,axiom,
! [F: nat > rule] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule ) ) ) ).
% bijI
thf(fact_1050_bijI,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ) ).
% bijI
thf(fact_1051_bijI,axiom,
! [F: nat > rat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat ) ) ) ).
% bijI
thf(fact_1052_bijI,axiom,
! [F: rat > rule] :
( ( inj_on_rat_rule @ F @ top_top_set_rat )
=> ( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule ) ) ) ).
% bijI
thf(fact_1053_bijI,axiom,
! [F: rat > nat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat ) ) ) ).
% bijI
thf(fact_1054_bijI,axiom,
! [F: rat > rat] :
( ( inj_on_rat_rat @ F @ top_top_set_rat )
=> ( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat ) ) ) ).
% bijI
thf(fact_1055_bijI,axiom,
! [F: rule > set_rat] :
( ( inj_on_rule_set_rat @ F @ top_top_set_rule )
=> ( ( ( image_rule_set_rat @ F @ top_top_set_rule )
= top_top_set_set_rat )
=> ( bij_be5592134826634264011et_rat @ F @ top_top_set_rule @ top_top_set_set_rat ) ) ) ).
% bijI
thf(fact_1056_bij__image__Compl__eq,axiom,
! [F: rule > rule,A: set_rule] :
( ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule )
=> ( ( image_rule_rule @ F @ ( uminus4869265918275750596t_rule @ A ) )
= ( uminus4869265918275750596t_rule @ ( image_rule_rule @ F @ A ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1057_bij__image__Compl__eq,axiom,
! [F: rule > nat,A: set_rule] :
( ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat )
=> ( ( image_rule_nat @ F @ ( uminus4869265918275750596t_rule @ A ) )
= ( uminus5710092332889474511et_nat @ ( image_rule_nat @ F @ A ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1058_bij__image__Compl__eq,axiom,
! [F: rule > rat,A: set_rule] :
( ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat )
=> ( ( image_rule_rat @ F @ ( uminus4869265918275750596t_rule @ A ) )
= ( uminus2201863774496077783et_rat @ ( image_rule_rat @ F @ A ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1059_bij__image__Compl__eq,axiom,
! [F: nat > rule,A: set_nat] :
( ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule )
=> ( ( image_nat_rule @ F @ ( uminus5710092332889474511et_nat @ A ) )
= ( uminus4869265918275750596t_rule @ ( image_nat_rule @ F @ A ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1060_bij__image__Compl__eq,axiom,
! [F: nat > nat,A: set_nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) )
= ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1061_bij__image__Compl__eq,axiom,
! [F: nat > rat,A: set_nat] :
( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
=> ( ( image_nat_rat @ F @ ( uminus5710092332889474511et_nat @ A ) )
= ( uminus2201863774496077783et_rat @ ( image_nat_rat @ F @ A ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1062_bij__image__Compl__eq,axiom,
! [F: rat > rule,A: set_rat] :
( ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule )
=> ( ( image_rat_rule @ F @ ( uminus2201863774496077783et_rat @ A ) )
= ( uminus4869265918275750596t_rule @ ( image_rat_rule @ F @ A ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1063_bij__image__Compl__eq,axiom,
! [F: rat > nat,A: set_rat] :
( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
=> ( ( image_rat_nat @ F @ ( uminus2201863774496077783et_rat @ A ) )
= ( uminus5710092332889474511et_nat @ ( image_rat_nat @ F @ A ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1064_bij__image__Compl__eq,axiom,
! [F: rat > rat,A: set_rat] :
( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
=> ( ( image_rat_rat @ F @ ( uminus2201863774496077783et_rat @ A ) )
= ( uminus2201863774496077783et_rat @ ( image_rat_rat @ F @ A ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1065_bij__image__Compl__eq,axiom,
! [F: rule > set_rat,A: set_rule] :
( ( bij_be5592134826634264011et_rat @ F @ top_top_set_rule @ top_top_set_set_rat )
=> ( ( image_rule_set_rat @ F @ ( uminus4869265918275750596t_rule @ A ) )
= ( uminus3456807040561714317et_rat @ ( image_rule_set_rat @ F @ A ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1066_bij__betw__inv__into__subset,axiom,
! [F: nat > rule,A: set_nat,A7: set_rule,B2: set_nat,B7: set_rule] :
( ( bij_betw_nat_rule @ F @ A @ A7 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ( image_nat_rule @ F @ B2 )
= B7 )
=> ( bij_betw_rule_nat @ ( hilber8541579349336805475t_rule @ A @ F ) @ B7 @ B2 ) ) ) ) ).
% bij_betw_inv_into_subset
thf(fact_1067_bij__betw__inv__into__subset,axiom,
! [F: nat > rat,A: set_nat,A7: set_rat,B2: set_nat,B7: set_rat] :
( ( bij_betw_nat_rat @ F @ A @ A7 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ( image_nat_rat @ F @ B2 )
= B7 )
=> ( bij_betw_rat_nat @ ( hilber2998747136712319222at_rat @ A @ F ) @ B7 @ B2 ) ) ) ) ).
% bij_betw_inv_into_subset
thf(fact_1068_bijection_Osurj,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule ) ) ).
% bijection.surj
thf(fact_1069_bijection_Osurj,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj
thf(fact_1070_bijection_Osurj,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat ) ) ).
% bijection.surj
thf(fact_1071_bijection_Osurj,axiom,
! [F: set_rat > set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( ( image_3939399684171694371et_rat @ F @ top_top_set_set_rat )
= top_top_set_set_rat ) ) ).
% bijection.surj
thf(fact_1072_bijection_Osurj,axiom,
! [F: set_nat > set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat )
= top_top_set_set_nat ) ) ).
% bijection.surj
thf(fact_1073_bijection_Osurj,axiom,
! [F: set_rule > set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( ( image_2455769455774476541t_rule @ F @ top_top_set_set_rule )
= top_top_set_set_rule ) ) ).
% bijection.surj
thf(fact_1074_bijection_Oinj,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( inj_on_rule_rule @ F @ top_top_set_rule ) ) ).
% bijection.inj
thf(fact_1075_bijection_Oinj,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% bijection.inj
thf(fact_1076_bijection_Oinj,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( inj_on_rat_rat @ F @ top_top_set_rat ) ) ).
% bijection.inj
thf(fact_1077_bijection_Oinj,axiom,
! [F: set_rat > set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( inj_on626919071704544911et_rat @ F @ top_top_set_set_rat ) ) ).
% bijection.inj
thf(fact_1078_bijection_Oinj,axiom,
! [F: set_nat > set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( inj_on4604407203859583615et_nat @ F @ top_top_set_set_nat ) ) ).
% bijection.inj
thf(fact_1079_bijection_Oinj,axiom,
! [F: set_rule > set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( inj_on733622247606601321t_rule @ F @ top_top_set_set_rule ) ) ).
% bijection.inj
thf(fact_1080_bijection_Oeq__invI,axiom,
! [F: rule > rule,A2: rule,B: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ A2 )
= ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ B ) )
=> ( A2 = B ) ) ) ).
% bijection.eq_invI
thf(fact_1081_bijection_Oeq__invI,axiom,
! [F: nat > nat,A2: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 )
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ B ) )
=> ( A2 = B ) ) ) ).
% bijection.eq_invI
thf(fact_1082_bijection_Oeq__invI,axiom,
! [F: rat > rat,A2: rat,B: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ A2 )
= ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ B ) )
=> ( A2 = B ) ) ) ).
% bijection.eq_invI
thf(fact_1083_bijection_Oeq__invI,axiom,
! [F: set_rat > set_rat,A2: set_rat,B: set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( ( ( hilber3461591988630719082et_rat @ top_top_set_set_rat @ F @ A2 )
= ( hilber3461591988630719082et_rat @ top_top_set_set_rat @ F @ B ) )
=> ( A2 = B ) ) ) ).
% bijection.eq_invI
thf(fact_1084_bijection_Oeq__invI,axiom,
! [F: set_nat > set_nat,A2: set_nat,B: set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( ( ( hilber7439080120785757786et_nat @ top_top_set_set_nat @ F @ A2 )
= ( hilber7439080120785757786et_nat @ top_top_set_set_nat @ F @ B ) )
=> ( A2 = B ) ) ) ).
% bijection.eq_invI
thf(fact_1085_bijection_Oeq__invI,axiom,
! [F: set_rule > set_rule,A2: set_rule,B: set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( ( ( hilber824017158074262596t_rule @ top_top_set_set_rule @ F @ A2 )
= ( hilber824017158074262596t_rule @ top_top_set_set_rule @ F @ B ) )
=> ( A2 = B ) ) ) ).
% bijection.eq_invI
thf(fact_1086_bijection_Oinv__left,axiom,
! [F: rule > rule,A2: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ ( F @ A2 ) )
= A2 ) ) ).
% bijection.inv_left
thf(fact_1087_bijection_Oinv__left,axiom,
! [F: nat > nat,A2: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ ( F @ A2 ) )
= A2 ) ) ).
% bijection.inv_left
thf(fact_1088_bijection_Oinv__left,axiom,
! [F: rat > rat,A2: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ ( F @ A2 ) )
= A2 ) ) ).
% bijection.inv_left
thf(fact_1089_bijection_Oinv__left,axiom,
! [F: set_rat > set_rat,A2: set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( ( hilber3461591988630719082et_rat @ top_top_set_set_rat @ F @ ( F @ A2 ) )
= A2 ) ) ).
% bijection.inv_left
thf(fact_1090_bijection_Oinv__left,axiom,
! [F: set_nat > set_nat,A2: set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( ( hilber7439080120785757786et_nat @ top_top_set_set_nat @ F @ ( F @ A2 ) )
= A2 ) ) ).
% bijection.inv_left
thf(fact_1091_bijection_Oinv__left,axiom,
! [F: set_rule > set_rule,A2: set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( ( hilber824017158074262596t_rule @ top_top_set_set_rule @ F @ ( F @ A2 ) )
= A2 ) ) ).
% bijection.inv_left
thf(fact_1092_bijection_Oinv__right,axiom,
! [F: rule > rule,A2: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( F @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ A2 ) )
= A2 ) ) ).
% bijection.inv_right
thf(fact_1093_bijection_Oinv__right,axiom,
! [F: nat > nat,A2: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 ) )
= A2 ) ) ).
% bijection.inv_right
thf(fact_1094_bijection_Oinv__right,axiom,
! [F: rat > rat,A2: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( F @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ A2 ) )
= A2 ) ) ).
% bijection.inv_right
thf(fact_1095_bijection_Oinv__right,axiom,
! [F: set_rat > set_rat,A2: set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( ( F @ ( hilber3461591988630719082et_rat @ top_top_set_set_rat @ F @ A2 ) )
= A2 ) ) ).
% bijection.inv_right
thf(fact_1096_bijection_Oinv__right,axiom,
! [F: set_nat > set_nat,A2: set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( ( F @ ( hilber7439080120785757786et_nat @ top_top_set_set_nat @ F @ A2 ) )
= A2 ) ) ).
% bijection.inv_right
thf(fact_1097_bijection_Oinv__right,axiom,
! [F: set_rule > set_rule,A2: set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( ( F @ ( hilber824017158074262596t_rule @ top_top_set_set_rule @ F @ A2 ) )
= A2 ) ) ).
% bijection.inv_right
thf(fact_1098_bijection_Oeq__inv__iff,axiom,
! [F: rule > rule,A2: rule,B: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ A2 )
= ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ B ) )
= ( A2 = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_1099_bijection_Oeq__inv__iff,axiom,
! [F: nat > nat,A2: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 )
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ B ) )
= ( A2 = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_1100_bijection_Oeq__inv__iff,axiom,
! [F: rat > rat,A2: rat,B: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ A2 )
= ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ B ) )
= ( A2 = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_1101_bijection_Oeq__inv__iff,axiom,
! [F: set_rat > set_rat,A2: set_rat,B: set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( ( ( hilber3461591988630719082et_rat @ top_top_set_set_rat @ F @ A2 )
= ( hilber3461591988630719082et_rat @ top_top_set_set_rat @ F @ B ) )
= ( A2 = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_1102_bijection_Oeq__inv__iff,axiom,
! [F: set_nat > set_nat,A2: set_nat,B: set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( ( ( hilber7439080120785757786et_nat @ top_top_set_set_nat @ F @ A2 )
= ( hilber7439080120785757786et_nat @ top_top_set_set_nat @ F @ B ) )
= ( A2 = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_1103_bijection_Oeq__inv__iff,axiom,
! [F: set_rule > set_rule,A2: set_rule,B: set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( ( ( hilber824017158074262596t_rule @ top_top_set_set_rule @ F @ A2 )
= ( hilber824017158074262596t_rule @ top_top_set_set_rule @ F @ B ) )
= ( A2 = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_1104_bijection_Oinv__left__eq__iff,axiom,
! [F: rule > rule,A2: rule,B: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ A2 )
= B )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_1105_bijection_Oinv__left__eq__iff,axiom,
! [F: nat > nat,A2: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 )
= B )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_1106_bijection_Oinv__left__eq__iff,axiom,
! [F: rat > rat,A2: rat,B: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ A2 )
= B )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_1107_bijection_Oinv__left__eq__iff,axiom,
! [F: set_rat > set_rat,A2: set_rat,B: set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( ( ( hilber3461591988630719082et_rat @ top_top_set_set_rat @ F @ A2 )
= B )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_1108_bijection_Oinv__left__eq__iff,axiom,
! [F: set_nat > set_nat,A2: set_nat,B: set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( ( ( hilber7439080120785757786et_nat @ top_top_set_set_nat @ F @ A2 )
= B )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_1109_bijection_Oinv__left__eq__iff,axiom,
! [F: set_rule > set_rule,A2: set_rule,B: set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( ( ( hilber824017158074262596t_rule @ top_top_set_set_rule @ F @ A2 )
= B )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_1110_bijection_Oinv__right__eq__iff,axiom,
! [F: rule > rule,B: rule,A2: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( B
= ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ A2 ) )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_1111_bijection_Oinv__right__eq__iff,axiom,
! [F: nat > nat,B: nat,A2: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( B
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 ) )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_1112_bijection_Oinv__right__eq__iff,axiom,
! [F: rat > rat,B: rat,A2: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( B
= ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ A2 ) )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_1113_bijection_Oinv__right__eq__iff,axiom,
! [F: set_rat > set_rat,B: set_rat,A2: set_rat] :
( ( hilber3240988290065685599et_rat @ F )
=> ( ( B
= ( hilber3461591988630719082et_rat @ top_top_set_set_rat @ F @ A2 ) )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_1114_bijection_Oinv__right__eq__iff,axiom,
! [F: set_nat > set_nat,B: set_nat,A2: set_nat] :
( ( hilber6749216848459082327et_nat @ F )
=> ( ( B
= ( hilber7439080120785757786et_nat @ top_top_set_set_nat @ F @ A2 ) )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_1115_bijection_Oinv__right__eq__iff,axiom,
! [F: set_rule > set_rule,B: set_rule,A2: set_rule] :
( ( hilber5661084989754287948t_rule @ F )
=> ( ( B
= ( hilber824017158074262596t_rule @ top_top_set_set_rule @ F @ A2 ) )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_1116_Ici__subset__Ioi__iff,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ A2 ) @ ( set_or1210151606488870762an_nat @ B ) )
= ( ord_less_nat @ B @ A2 ) ) ).
% Ici_subset_Ioi_iff
thf(fact_1117_Schroeder__Bernstein,axiom,
! [F: rule > nat,A: set_rule,B2: set_nat,G: nat > rule] :
( ( inj_on_rule_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_rule_nat @ F @ A ) @ B2 )
=> ( ( inj_on_nat_rule @ G @ B2 )
=> ( ( ord_less_eq_set_rule @ ( image_nat_rule @ G @ B2 ) @ A )
=> ? [H3: rule > nat] : ( bij_betw_rule_nat @ H3 @ A @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_1118_Schroeder__Bernstein,axiom,
! [F: rat > nat,A: set_rat,B2: set_nat,G: nat > rat] :
( ( inj_on_rat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_rat_nat @ F @ A ) @ B2 )
=> ( ( inj_on_nat_rat @ G @ B2 )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ G @ B2 ) @ A )
=> ? [H3: rat > nat] : ( bij_betw_rat_nat @ H3 @ A @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_1119_Schroeder__Bernstein,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule,G: rule > nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ B2 )
=> ( ( inj_on_rule_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_rule_nat @ G @ B2 ) @ A )
=> ? [H3: nat > rule] : ( bij_betw_nat_rule @ H3 @ A @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_1120_Schroeder__Bernstein,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat,G: rat > nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ B2 )
=> ( ( inj_on_rat_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_rat_nat @ G @ B2 ) @ A )
=> ? [H3: nat > rat] : ( bij_betw_nat_rat @ H3 @ A @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_1121_bijI_H,axiom,
! [F: rule > rule] :
( ! [X2: rule,Y5: rule] :
( ( ( F @ X2 )
= ( F @ Y5 ) )
= ( X2 = Y5 ) )
=> ( ! [Y5: rule] :
? [X6: rule] :
( Y5
= ( F @ X6 ) )
=> ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule ) ) ) ).
% bijI'
thf(fact_1122_bijI_H,axiom,
! [F: rule > nat] :
( ! [X2: rule,Y5: rule] :
( ( ( F @ X2 )
= ( F @ Y5 ) )
= ( X2 = Y5 ) )
=> ( ! [Y5: nat] :
? [X6: rule] :
( Y5
= ( F @ X6 ) )
=> ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat ) ) ) ).
% bijI'
thf(fact_1123_bijI_H,axiom,
! [F: rule > rat] :
( ! [X2: rule,Y5: rule] :
( ( ( F @ X2 )
= ( F @ Y5 ) )
= ( X2 = Y5 ) )
=> ( ! [Y5: rat] :
? [X6: rule] :
( Y5
= ( F @ X6 ) )
=> ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat ) ) ) ).
% bijI'
thf(fact_1124_bijI_H,axiom,
! [F: nat > rule] :
( ! [X2: nat,Y5: nat] :
( ( ( F @ X2 )
= ( F @ Y5 ) )
= ( X2 = Y5 ) )
=> ( ! [Y5: rule] :
? [X6: nat] :
( Y5
= ( F @ X6 ) )
=> ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule ) ) ) ).
% bijI'
thf(fact_1125_bijI_H,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y5: nat] :
( ( ( F @ X2 )
= ( F @ Y5 ) )
= ( X2 = Y5 ) )
=> ( ! [Y5: nat] :
? [X6: nat] :
( Y5
= ( F @ X6 ) )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ) ).
% bijI'
thf(fact_1126_bijI_H,axiom,
! [F: nat > rat] :
( ! [X2: nat,Y5: nat] :
( ( ( F @ X2 )
= ( F @ Y5 ) )
= ( X2 = Y5 ) )
=> ( ! [Y5: rat] :
? [X6: nat] :
( Y5
= ( F @ X6 ) )
=> ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat ) ) ) ).
% bijI'
thf(fact_1127_bijI_H,axiom,
! [F: rat > rule] :
( ! [X2: rat,Y5: rat] :
( ( ( F @ X2 )
= ( F @ Y5 ) )
= ( X2 = Y5 ) )
=> ( ! [Y5: rule] :
? [X6: rat] :
( Y5
= ( F @ X6 ) )
=> ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule ) ) ) ).
% bijI'
thf(fact_1128_bijI_H,axiom,
! [F: rat > nat] :
( ! [X2: rat,Y5: rat] :
( ( ( F @ X2 )
= ( F @ Y5 ) )
= ( X2 = Y5 ) )
=> ( ! [Y5: nat] :
? [X6: rat] :
( Y5
= ( F @ X6 ) )
=> ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat ) ) ) ).
% bijI'
thf(fact_1129_bijI_H,axiom,
! [F: rat > rat] :
( ! [X2: rat,Y5: rat] :
( ( ( F @ X2 )
= ( F @ Y5 ) )
= ( X2 = Y5 ) )
=> ( ! [Y5: rat] :
? [X6: rat] :
( Y5
= ( F @ X6 ) )
=> ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat ) ) ) ).
% bijI'
thf(fact_1130_bijI_H,axiom,
! [F: rule > set_rat] :
( ! [X2: rule,Y5: rule] :
( ( ( F @ X2 )
= ( F @ Y5 ) )
= ( X2 = Y5 ) )
=> ( ! [Y5: set_rat] :
? [X6: rule] :
( Y5
= ( F @ X6 ) )
=> ( bij_be5592134826634264011et_rat @ F @ top_top_set_rule @ top_top_set_set_rat ) ) ) ).
% bijI'
thf(fact_1131_mono__image__least,axiom,
! [F: rat > nat,M3: rat,N3: rat,M4: nat,N4: nat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( ( image_rat_nat @ F @ ( set_or4029947393144176647an_rat @ M3 @ N3 ) )
= ( set_or4665077453230672383an_nat @ M4 @ N4 ) )
=> ( ( ord_less_rat @ M3 @ N3 )
=> ( ( F @ M3 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_1132_mono__image__least,axiom,
! [F: set_rat > nat,M3: set_rat,N3: set_rat,M4: nat,N4: nat] :
( ( monoto7746382625157022407at_nat @ top_top_set_set_rat @ ord_less_eq_set_rat @ ord_less_eq_nat @ F )
=> ( ( ( image_set_rat_nat @ F @ ( set_or32047845639629757et_rat @ M3 @ N3 ) )
= ( set_or4665077453230672383an_nat @ M4 @ N4 ) )
=> ( ( ord_less_set_rat @ M3 @ N3 )
=> ( ( F @ M3 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_1133_mono__image__least,axiom,
! [F: set_nat > nat,M3: set_nat,N3: set_nat,M4: nat,N4: nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( ( ( image_set_nat_nat @ F @ ( set_or3540276404033026485et_nat @ M3 @ N3 ) )
= ( set_or4665077453230672383an_nat @ M4 @ N4 ) )
=> ( ( ord_less_set_nat @ M3 @ N3 )
=> ( ( F @ M3 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_1134_mono__image__least,axiom,
! [F: set_rule > nat,M3: set_rule,N3: set_rule,M4: nat,N4: nat] :
( ( monoto593345095951663796le_nat @ top_top_set_set_rule @ ord_less_eq_set_rule @ ord_less_eq_nat @ F )
=> ( ( ( image_set_rule_nat @ F @ ( set_or4567957997179852970t_rule @ M3 @ N3 ) )
= ( set_or4665077453230672383an_nat @ M4 @ N4 ) )
=> ( ( ord_less_set_rule @ M3 @ N3 )
=> ( ( F @ M3 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_1135_mono__image__least,axiom,
! [F: nat > rat,M3: nat,N3: nat,M4: rat,N4: rat] :
( ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_rat @ F )
=> ( ( ( image_nat_rat @ F @ ( set_or4665077453230672383an_nat @ M3 @ N3 ) )
= ( set_or4029947393144176647an_rat @ M4 @ N4 ) )
=> ( ( ord_less_nat @ M3 @ N3 )
=> ( ( F @ M3 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_1136_mono__image__least,axiom,
! [F: nat > nat,M3: nat,N3: nat,M4: nat,N4: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ( image_nat_nat @ F @ ( set_or4665077453230672383an_nat @ M3 @ N3 ) )
= ( set_or4665077453230672383an_nat @ M4 @ N4 ) )
=> ( ( ord_less_nat @ M3 @ N3 )
=> ( ( F @ M3 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_1137_bdd__above__image__mono,axiom,
! [F: rat > nat,A: set_rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( condit1579696412822616692ve_rat @ A )
=> ( condit2214826472909112428ve_nat @ ( image_rat_nat @ F @ A ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1138_bdd__above__image__mono,axiom,
! [F: set_rat > nat,A: set_set_rat] :
( ( monoto7746382625157022407at_nat @ top_top_set_set_rat @ ord_less_eq_set_rat @ ord_less_eq_nat @ F )
=> ( ( condit1969311730731577898et_rat @ A )
=> ( condit2214826472909112428ve_nat @ ( image_set_rat_nat @ F @ A ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1139_bdd__above__image__mono,axiom,
! [F: set_nat > nat,A: set_set_nat] :
( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F )
=> ( ( condit5477540289124974626et_nat @ A )
=> ( condit2214826472909112428ve_nat @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1140_bdd__above__image__mono,axiom,
! [F: set_rule > nat,A: set_set_rule] :
( ( monoto593345095951663796le_nat @ top_top_set_set_rule @ ord_less_eq_set_rule @ ord_less_eq_nat @ F )
=> ( ( condit9083804844586111511t_rule @ A )
=> ( condit2214826472909112428ve_nat @ ( image_set_rule_nat @ F @ A ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1141_bdd__above__image__mono,axiom,
! [F: nat > rat,A: set_nat] :
( ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_rat @ F )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( condit1579696412822616692ve_rat @ ( image_nat_rat @ F @ A ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1142_bdd__above__image__mono,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1143_Inter__UNIV__conv_I2_J,axiom,
! [A: set_set_rule] :
( ( top_top_set_rule
= ( comple5773694076043965236t_rule @ A ) )
= ( ! [X3: set_rule] :
( ( member_set_rule @ X3 @ A )
=> ( X3 = top_top_set_rule ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_1144_Inter__UNIV__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ A ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( X3 = top_top_set_nat ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_1145_Inter__UNIV__conv_I2_J,axiom,
! [A: set_set_rat] :
( ( top_top_set_rat
= ( comple4298007329820168263et_rat @ A ) )
= ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
=> ( X3 = top_top_set_rat ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_1146_Inter__UNIV__conv_I2_J,axiom,
! [A: set_set_set_rat] :
( ( top_top_set_set_rat
= ( comple3908394330019556605et_rat @ A ) )
= ( ! [X3: set_set_rat] :
( ( member_set_set_rat @ X3 @ A )
=> ( X3 = top_top_set_set_rat ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_1147_Inter__UNIV__conv_I2_J,axiom,
! [A: set_set_set_nat] :
( ( top_top_set_set_nat
= ( comple1065008630642458357et_nat @ A ) )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( X3 = top_top_set_set_nat ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_1148_Inter__UNIV__conv_I2_J,axiom,
! [A: set_set_set_rule] :
( ( top_top_set_set_rule
= ( comple1145120538619166314t_rule @ A ) )
= ( ! [X3: set_set_rule] :
( ( member_set_set_rule @ X3 @ A )
=> ( X3 = top_top_set_set_rule ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_1149_Inter__UNIV__conv_I1_J,axiom,
! [A: set_set_rule] :
( ( ( comple5773694076043965236t_rule @ A )
= top_top_set_rule )
= ( ! [X3: set_rule] :
( ( member_set_rule @ X3 @ A )
=> ( X3 = top_top_set_rule ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_1150_Inter__UNIV__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7806235888213564991et_nat @ A )
= top_top_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( X3 = top_top_set_nat ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_1151_Inter__UNIV__conv_I1_J,axiom,
! [A: set_set_rat] :
( ( ( comple4298007329820168263et_rat @ A )
= top_top_set_rat )
= ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
=> ( X3 = top_top_set_rat ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_1152_Inter__UNIV__conv_I1_J,axiom,
! [A: set_set_set_rat] :
( ( ( comple3908394330019556605et_rat @ A )
= top_top_set_set_rat )
= ( ! [X3: set_set_rat] :
( ( member_set_set_rat @ X3 @ A )
=> ( X3 = top_top_set_set_rat ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_1153_Inter__UNIV__conv_I1_J,axiom,
! [A: set_set_set_nat] :
( ( ( comple1065008630642458357et_nat @ A )
= top_top_set_set_nat )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( X3 = top_top_set_set_nat ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_1154_Inter__UNIV__conv_I1_J,axiom,
! [A: set_set_set_rule] :
( ( ( comple1145120538619166314t_rule @ A )
= top_top_set_set_rule )
= ( ! [X3: set_set_rule] :
( ( member_set_set_rule @ X3 @ A )
=> ( X3 = top_top_set_set_rule ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_1155_Inf__top__conv_I2_J,axiom,
! [A: set_set_rule] :
( ( top_top_set_rule
= ( comple5773694076043965236t_rule @ A ) )
= ( ! [X3: set_rule] :
( ( member_set_rule @ X3 @ A )
=> ( X3 = top_top_set_rule ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_1156_Inf__top__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ A ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( X3 = top_top_set_nat ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_1157_Inf__top__conv_I2_J,axiom,
! [A: set_set_rat] :
( ( top_top_set_rat
= ( comple4298007329820168263et_rat @ A ) )
= ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
=> ( X3 = top_top_set_rat ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_1158_Inf__top__conv_I2_J,axiom,
! [A: set_set_set_rat] :
( ( top_top_set_set_rat
= ( comple3908394330019556605et_rat @ A ) )
= ( ! [X3: set_set_rat] :
( ( member_set_set_rat @ X3 @ A )
=> ( X3 = top_top_set_set_rat ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_1159_Inf__top__conv_I2_J,axiom,
! [A: set_set_set_nat] :
( ( top_top_set_set_nat
= ( comple1065008630642458357et_nat @ A ) )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( X3 = top_top_set_set_nat ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_1160_Inf__top__conv_I2_J,axiom,
! [A: set_set_set_rule] :
( ( top_top_set_set_rule
= ( comple1145120538619166314t_rule @ A ) )
= ( ! [X3: set_set_rule] :
( ( member_set_set_rule @ X3 @ A )
=> ( X3 = top_top_set_set_rule ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_1161_Inf__top__conv_I1_J,axiom,
! [A: set_set_rule] :
( ( ( comple5773694076043965236t_rule @ A )
= top_top_set_rule )
= ( ! [X3: set_rule] :
( ( member_set_rule @ X3 @ A )
=> ( X3 = top_top_set_rule ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_1162_Inf__top__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7806235888213564991et_nat @ A )
= top_top_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( X3 = top_top_set_nat ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_1163_Inf__top__conv_I1_J,axiom,
! [A: set_set_rat] :
( ( ( comple4298007329820168263et_rat @ A )
= top_top_set_rat )
= ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
=> ( X3 = top_top_set_rat ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_1164_Inf__top__conv_I1_J,axiom,
! [A: set_set_set_rat] :
( ( ( comple3908394330019556605et_rat @ A )
= top_top_set_set_rat )
= ( ! [X3: set_set_rat] :
( ( member_set_set_rat @ X3 @ A )
=> ( X3 = top_top_set_set_rat ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_1165_Inf__top__conv_I1_J,axiom,
! [A: set_set_set_nat] :
( ( ( comple1065008630642458357et_nat @ A )
= top_top_set_set_nat )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( X3 = top_top_set_set_nat ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_1166_Inf__top__conv_I1_J,axiom,
! [A: set_set_set_rule] :
( ( ( comple1145120538619166314t_rule @ A )
= top_top_set_set_rule )
= ( ! [X3: set_set_rule] :
( ( member_set_set_rule @ X3 @ A )
=> ( X3 = top_top_set_set_rule ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_1167_bdd__above_OI,axiom,
! [A: set_rat,M2: rat] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_rat @ X2 @ M2 ) )
=> ( condit1579696412822616692ve_rat @ A ) ) ).
% bdd_above.I
thf(fact_1168_bdd__above_OI,axiom,
! [A: set_nat,M2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ( condit2214826472909112428ve_nat @ A ) ) ).
% bdd_above.I
thf(fact_1169_bdd__above__Iic,axiom,
! [B: set_rat] : ( condit1969311730731577898et_rat @ ( set_or728397472755099399et_rat @ B ) ) ).
% bdd_above_Iic
thf(fact_1170_bdd__above__Iic,axiom,
! [B: set_nat] : ( condit5477540289124974626et_nat @ ( set_or4236626031148496127et_nat @ B ) ) ).
% bdd_above_Iic
thf(fact_1171_bdd__above__Iic,axiom,
! [B: set_rule] : ( condit9083804844586111511t_rule @ ( set_or7438564873865873908t_rule @ B ) ) ).
% bdd_above_Iic
thf(fact_1172_atLeastLessThan__iff,axiom,
! [I: rat,L: rat,U: rat] :
( ( member_rat @ I @ ( set_or4029947393144176647an_rat @ L @ U ) )
= ( ( ord_less_eq_rat @ L @ I )
& ( ord_less_rat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1173_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1174_ivl__subset,axiom,
! [I: nat,J2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J2 ) @ ( set_or4665077453230672383an_nat @ M3 @ N3 ) )
= ( ( ord_less_eq_nat @ J2 @ I )
| ( ( ord_less_eq_nat @ M3 @ I )
& ( ord_less_eq_nat @ J2 @ N3 ) ) ) ) ).
% ivl_subset
thf(fact_1175_cInf__atLeastLessThan,axiom,
! [Y: nat,X: nat] :
( ( ord_less_nat @ Y @ X )
=> ( ( complete_Inf_Inf_nat @ ( set_or4665077453230672383an_nat @ Y @ X ) )
= Y ) ) ).
% cInf_atLeastLessThan
thf(fact_1176_ivl__diff,axiom,
! [I: nat,N3: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M3 ) @ ( set_or4665077453230672383an_nat @ I @ N3 ) )
= ( set_or4665077453230672383an_nat @ N3 @ M3 ) ) ) ).
% ivl_diff
thf(fact_1177_atLeastLessThan__eq__iff,axiom,
! [A2: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D3 ) )
= ( ( A2 = C2 )
& ( B = D3 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_1178_Ico__eq__Ico,axiom,
! [L: nat,H2: nat,L2: nat,H: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H2 )
= ( set_or4665077453230672383an_nat @ L2 @ H ) )
= ( ( ( L = L2 )
& ( H2 = H ) )
| ( ~ ( ord_less_nat @ L @ H2 )
& ~ ( ord_less_nat @ L2 @ H ) ) ) ) ).
% Ico_eq_Ico
thf(fact_1179_atLeastLessThan__inj_I1_J,axiom,
! [A2: nat,B: nat,C2: nat,D3: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D3 ) )
=> ( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( A2 = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_1180_atLeastLessThan__inj_I2_J,axiom,
! [A2: nat,B: nat,C2: nat,D3: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D3 ) )
=> ( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( B = D3 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_1181_cInf__eq__minimum,axiom,
! [Z2: nat,X4: set_nat] :
( ( member_nat @ Z2 @ X4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ( ( complete_Inf_Inf_nat @ X4 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_1182_cInf__eq,axiom,
! [X4: set_nat,A2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_eq_nat @ A2 @ X2 ) )
=> ( ! [Y5: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ X4 )
=> ( ord_less_eq_nat @ Y5 @ X6 ) )
=> ( ord_less_eq_nat @ Y5 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X4 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_1183_wellorder__Inf__le1,axiom,
! [K: nat,A: set_nat] :
( ( member_nat @ K @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ K ) ) ).
% wellorder_Inf_le1
thf(fact_1184_bdd__above_OE,axiom,
! [A: set_rat] :
( ( condit1579696412822616692ve_rat @ A )
=> ~ ! [M5: rat] :
~ ! [X6: rat] :
( ( member_rat @ X6 @ A )
=> ( ord_less_eq_rat @ X6 @ M5 ) ) ) ).
% bdd_above.E
thf(fact_1185_bdd__above_OE,axiom,
! [A: set_nat] :
( ( condit2214826472909112428ve_nat @ A )
=> ~ ! [M5: nat] :
~ ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( ord_less_eq_nat @ X6 @ M5 ) ) ) ).
% bdd_above.E
thf(fact_1186_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A5: set_nat] :
? [M6: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( ord_less_eq_nat @ X3 @ M6 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1187_Inf__INT__eq,axiom,
( comple2477142665972227838_rat_o
= ( ^ [S3: set_rat_o,X3: rat] : ( member_rat @ X3 @ ( comple4298007329820168263et_rat @ ( image_rat_o_set_rat @ collect_rat @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1188_Inf__INT__eq,axiom,
( comple6214475593288795910_nat_o
= ( ^ [S3: set_nat_o,X3: nat] : ( member_nat @ X3 @ ( comple7806235888213564991et_nat @ ( image_nat_o_set_nat @ collect_nat @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1189_Inf__INT__eq,axiom,
( comple715424409190658129rule_o
= ( ^ [S3: set_rule_o,X3: rule] : ( member_rule @ X3 @ ( comple5773694076043965236t_rule @ ( image_1281159361656534528t_rule @ collect_rule @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1190_wellorder__InfI,axiom,
! [K: nat,A: set_nat] :
( ( member_nat @ K @ A )
=> ( member_nat @ ( complete_Inf_Inf_nat @ A ) @ A ) ) ).
% wellorder_InfI
thf(fact_1191_atLeastLessThan__subset__iff,axiom,
! [A2: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A2 @ B ) @ ( set_or4665077453230672383an_nat @ C2 @ D3 ) )
=> ( ( ord_less_eq_nat @ B @ A2 )
| ( ( ord_less_eq_nat @ C2 @ A2 )
& ( ord_less_eq_nat @ B @ D3 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1192_bdd__above_OI2,axiom,
! [A: set_nat,F: nat > rat,M2: rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ M2 ) )
=> ( condit1579696412822616692ve_rat @ ( image_nat_rat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1193_bdd__above_OI2,axiom,
! [A: set_rat,F: rat > nat,M2: nat] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M2 ) )
=> ( condit2214826472909112428ve_nat @ ( image_rat_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1194_bdd__above_OI2,axiom,
! [A: set_nat,F: nat > nat,M2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M2 ) )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1195_bdd__above_OI2,axiom,
! [A: set_rule,F: rule > nat,M2: nat] :
( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M2 ) )
=> ( condit2214826472909112428ve_nat @ ( image_rule_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1196_cSup__upper,axiom,
! [X: nat,X4: set_nat] :
( ( member_nat @ X @ X4 )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ).
% cSup_upper
thf(fact_1197_cSup__upper,axiom,
! [X: set_rat,X4: set_set_rat] :
( ( member_set_rat @ X @ X4 )
=> ( ( condit1969311730731577898et_rat @ X4 )
=> ( ord_less_eq_set_rat @ X @ ( comple3890839924845867745et_rat @ X4 ) ) ) ) ).
% cSup_upper
thf(fact_1198_cSup__upper,axiom,
! [X: set_nat,X4: set_set_nat] :
( ( member_set_nat @ X @ X4 )
=> ( ( condit5477540289124974626et_nat @ X4 )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ).
% cSup_upper
thf(fact_1199_cSup__upper,axiom,
! [X: set_rule,X4: set_set_rule] :
( ( member_set_rule @ X @ X4 )
=> ( ( condit9083804844586111511t_rule @ X4 )
=> ( ord_less_eq_set_rule @ X @ ( comple2146307154184993742t_rule @ X4 ) ) ) ) ).
% cSup_upper
thf(fact_1200_cSup__upper2,axiom,
! [X: nat,X4: set_nat,Y: nat] :
( ( member_nat @ X @ X4 )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ord_less_eq_nat @ Y @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1201_cSup__upper2,axiom,
! [X: set_rat,X4: set_set_rat,Y: set_rat] :
( ( member_set_rat @ X @ X4 )
=> ( ( ord_less_eq_set_rat @ Y @ X )
=> ( ( condit1969311730731577898et_rat @ X4 )
=> ( ord_less_eq_set_rat @ Y @ ( comple3890839924845867745et_rat @ X4 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1202_cSup__upper2,axiom,
! [X: set_nat,X4: set_set_nat,Y: set_nat] :
( ( member_set_nat @ X @ X4 )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( condit5477540289124974626et_nat @ X4 )
=> ( ord_less_eq_set_nat @ Y @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1203_cSup__upper2,axiom,
! [X: set_rule,X4: set_set_rule,Y: set_rule] :
( ( member_set_rule @ X @ X4 )
=> ( ( ord_less_eq_set_rule @ Y @ X )
=> ( ( condit9083804844586111511t_rule @ X4 )
=> ( ord_less_eq_set_rule @ Y @ ( comple2146307154184993742t_rule @ X4 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1204_cSUP__upper,axiom,
! [X: rat,A: set_rat,F: rat > nat] :
( ( member_rat @ X @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_rat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( complete_Sup_Sup_nat @ ( image_rat_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1205_cSUP__upper,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1206_cSUP__upper,axiom,
! [X: rule,A: set_rule,F: rule > nat] :
( ( member_rule @ X @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_rule_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( complete_Sup_Sup_nat @ ( image_rule_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1207_cSUP__upper,axiom,
! [X: rat,A: set_rat,F: rat > set_rat] :
( ( member_rat @ X @ A )
=> ( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ F @ A ) )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1208_cSUP__upper,axiom,
! [X: nat,A: set_nat,F: nat > set_rat] :
( ( member_nat @ X @ A )
=> ( ( condit1969311730731577898et_rat @ ( image_nat_set_rat @ F @ A ) )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1209_cSUP__upper,axiom,
! [X: rule,A: set_rule,F: rule > set_rat] :
( ( member_rule @ X @ A )
=> ( ( condit1969311730731577898et_rat @ ( image_rule_set_rat @ F @ A ) )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1210_cSUP__upper,axiom,
! [X: rat,A: set_rat,F: rat > set_nat] :
( ( member_rat @ X @ A )
=> ( ( condit5477540289124974626et_nat @ ( image_rat_set_nat @ F @ A ) )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1211_cSUP__upper,axiom,
! [X: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A ) )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1212_cSUP__upper,axiom,
! [X: rule,A: set_rule,F: rule > set_nat] :
( ( member_rule @ X @ A )
=> ( ( condit5477540289124974626et_nat @ ( image_rule_set_nat @ F @ A ) )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1213_cSUP__upper,axiom,
! [X: rat,A: set_rat,F: rat > set_rule] :
( ( member_rat @ X @ A )
=> ( ( condit9083804844586111511t_rule @ ( image_rat_set_rule @ F @ A ) )
=> ( ord_less_eq_set_rule @ ( F @ X ) @ ( comple2146307154184993742t_rule @ ( image_rat_set_rule @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1214_cSUP__upper2,axiom,
! [F: rat > nat,A: set_rat,X: rat,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_rat_nat @ F @ A ) )
=> ( ( member_rat @ X @ A )
=> ( ( ord_less_eq_nat @ U @ ( F @ X ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_rat_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1215_cSUP__upper2,axiom,
! [F: nat > nat,A: set_nat,X: nat,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( member_nat @ X @ A )
=> ( ( ord_less_eq_nat @ U @ ( F @ X ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1216_cSUP__upper2,axiom,
! [F: rule > nat,A: set_rule,X: rule,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_rule_nat @ F @ A ) )
=> ( ( member_rule @ X @ A )
=> ( ( ord_less_eq_nat @ U @ ( F @ X ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_rule_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1217_cSUP__upper2,axiom,
! [F: rat > set_rat,A: set_rat,X: rat,U: set_rat] :
( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ F @ A ) )
=> ( ( member_rat @ X @ A )
=> ( ( ord_less_eq_set_rat @ U @ ( F @ X ) )
=> ( ord_less_eq_set_rat @ U @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1218_cSUP__upper2,axiom,
! [F: nat > set_rat,A: set_nat,X: nat,U: set_rat] :
( ( condit1969311730731577898et_rat @ ( image_nat_set_rat @ F @ A ) )
=> ( ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_rat @ U @ ( F @ X ) )
=> ( ord_less_eq_set_rat @ U @ ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1219_cSUP__upper2,axiom,
! [F: rule > set_rat,A: set_rule,X: rule,U: set_rat] :
( ( condit1969311730731577898et_rat @ ( image_rule_set_rat @ F @ A ) )
=> ( ( member_rule @ X @ A )
=> ( ( ord_less_eq_set_rat @ U @ ( F @ X ) )
=> ( ord_less_eq_set_rat @ U @ ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1220_cSUP__upper2,axiom,
! [F: rat > set_nat,A: set_rat,X: rat,U: set_nat] :
( ( condit5477540289124974626et_nat @ ( image_rat_set_nat @ F @ A ) )
=> ( ( member_rat @ X @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ X ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1221_cSUP__upper2,axiom,
! [F: nat > set_nat,A: set_nat,X: nat,U: set_nat] :
( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A ) )
=> ( ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ X ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1222_cSUP__upper2,axiom,
! [F: rule > set_nat,A: set_rule,X: rule,U: set_nat] :
( ( condit5477540289124974626et_nat @ ( image_rule_set_nat @ F @ A ) )
=> ( ( member_rule @ X @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ X ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1223_cSUP__upper2,axiom,
! [F: rat > set_rule,A: set_rat,X: rat,U: set_rule] :
( ( condit9083804844586111511t_rule @ ( image_rat_set_rule @ F @ A ) )
=> ( ( member_rat @ X @ A )
=> ( ( ord_less_eq_set_rule @ U @ ( F @ X ) )
=> ( ord_less_eq_set_rule @ U @ ( comple2146307154184993742t_rule @ ( image_rat_set_rule @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1224_uminus__Inf,axiom,
! [A: set_set_rat] :
( ( uminus2201863774496077783et_rat @ ( comple4298007329820168263et_rat @ A ) )
= ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ uminus2201863774496077783et_rat @ A ) ) ) ).
% uminus_Inf
thf(fact_1225_uminus__Inf,axiom,
! [A: set_set_nat] :
( ( uminus5710092332889474511et_nat @ ( comple7806235888213564991et_nat @ A ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ uminus5710092332889474511et_nat @ A ) ) ) ).
% uminus_Inf
thf(fact_1226_uminus__Inf,axiom,
! [A: set_set_rule] :
( ( uminus4869265918275750596t_rule @ ( comple5773694076043965236t_rule @ A ) )
= ( comple2146307154184993742t_rule @ ( image_2455769455774476541t_rule @ uminus4869265918275750596t_rule @ A ) ) ) ).
% uminus_Inf
thf(fact_1227_uminus__Sup,axiom,
! [A: set_set_rat] :
( ( uminus2201863774496077783et_rat @ ( comple3890839924845867745et_rat @ A ) )
= ( comple4298007329820168263et_rat @ ( image_3939399684171694371et_rat @ uminus2201863774496077783et_rat @ A ) ) ) ).
% uminus_Sup
thf(fact_1228_uminus__Sup,axiom,
! [A: set_set_nat] :
( ( uminus5710092332889474511et_nat @ ( comple7399068483239264473et_nat @ A ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ uminus5710092332889474511et_nat @ A ) ) ) ).
% uminus_Sup
thf(fact_1229_uminus__Sup,axiom,
! [A: set_set_rule] :
( ( uminus4869265918275750596t_rule @ ( comple2146307154184993742t_rule @ A ) )
= ( comple5773694076043965236t_rule @ ( image_2455769455774476541t_rule @ uminus4869265918275750596t_rule @ A ) ) ) ).
% uminus_Sup
thf(fact_1230_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X9: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X9 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X9 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_1231_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N3 @ ( F @ N3 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1232_incseq__imp__monoseq,axiom,
! [X4: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X4 )
=> ( topolo4902158794631467389eq_nat @ X4 ) ) ).
% incseq_imp_monoseq
thf(fact_1233_cSUP__subset__mono,axiom,
! [A: set_rat,G: rat > nat,B2: set_rat,F: rat > nat] :
( ( A != bot_bot_set_rat )
=> ( ( condit2214826472909112428ve_nat @ ( image_rat_nat @ G @ B2 ) )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_rat_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_rat_nat @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1234_cSUP__subset__mono,axiom,
! [A: set_nat,G: nat > nat,B2: set_nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B2 ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [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 @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1235_cSUP__subset__mono,axiom,
! [A: set_rule,G: rule > nat,B2: set_rule,F: rule > nat] :
( ( A != bot_bot_set_rule )
=> ( ( condit2214826472909112428ve_nat @ ( image_rule_nat @ G @ B2 ) )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_rule_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_rule_nat @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1236_cSUP__subset__mono,axiom,
! [A: set_rat,G: rat > set_rat,B2: set_rat,F: rat > set_rat] :
( ( A != bot_bot_set_rat )
=> ( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ G @ B2 ) )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_set_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1237_cSUP__subset__mono,axiom,
! [A: set_nat,G: nat > set_rat,B2: set_nat,F: nat > set_rat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1969311730731577898et_rat @ ( image_nat_set_rat @ G @ B2 ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1238_cSUP__subset__mono,axiom,
! [A: set_rule,G: rule > set_rat,B2: set_rule,F: rule > set_rat] :
( ( A != bot_bot_set_rule )
=> ( ( condit1969311730731577898et_rat @ ( image_rule_set_rat @ G @ B2 ) )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( ord_less_eq_set_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_rule_set_rat @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1239_cSUP__subset__mono,axiom,
! [A: set_rat,G: rat > set_nat,B2: set_rat,F: rat > set_nat] :
( ( A != bot_bot_set_rat )
=> ( ( condit5477540289124974626et_nat @ ( image_rat_set_nat @ G @ B2 ) )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1240_cSUP__subset__mono,axiom,
! [A: set_nat,G: nat > set_nat,B2: set_nat,F: nat > set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B2 ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1241_cSUP__subset__mono,axiom,
! [A: set_rule,G: rule > set_nat,B2: set_rule,F: rule > set_nat] :
( ( A != bot_bot_set_rule )
=> ( ( condit5477540289124974626et_nat @ ( image_rule_set_nat @ G @ B2 ) )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1242_cSUP__subset__mono,axiom,
! [A: set_rat,G: rat > set_rule,B2: set_rat,F: rat > set_rule] :
( ( A != bot_bot_set_rat )
=> ( ( condit9083804844586111511t_rule @ ( image_rat_set_rule @ G @ B2 ) )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_set_rule @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_rule @ ( comple2146307154184993742t_rule @ ( image_rat_set_rule @ F @ A ) ) @ ( comple2146307154184993742t_rule @ ( image_rat_set_rule @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1243_gfp__ordinal__induct,axiom,
! [F: set_rat > set_rat,P: set_rat > $o] :
( ( monoto6994633993926870149et_rat @ top_top_set_set_rat @ ord_less_eq_set_rat @ ord_less_eq_set_rat @ F )
=> ( ! [S6: set_rat] :
( ( P @ S6 )
=> ( ( ord_less_eq_set_rat @ ( comple7311222267670308624et_rat @ F ) @ S6 )
=> ( P @ ( F @ S6 ) ) ) )
=> ( ! [M5: set_set_rat] :
( ! [X6: set_rat] :
( ( member_set_rat @ X6 @ M5 )
=> ( P @ X6 ) )
=> ( P @ ( comple4298007329820168263et_rat @ M5 ) ) )
=> ( P @ ( comple7311222267670308624et_rat @ F ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_1244_gfp__ordinal__induct,axiom,
! [F: set_nat > set_nat,P: set_nat > $o] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ! [S6: set_nat] :
( ( P @ S6 )
=> ( ( ord_less_eq_set_nat @ ( comple1596078789208929544et_nat @ F ) @ S6 )
=> ( P @ ( F @ S6 ) ) ) )
=> ( ! [M5: set_set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ M5 )
=> ( P @ X6 ) )
=> ( P @ ( comple7806235888213564991et_nat @ M5 ) ) )
=> ( P @ ( comple1596078789208929544et_nat @ F ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_1245_gfp__ordinal__induct,axiom,
! [F: set_rule > set_rule,P: set_rule > $o] :
( ( monoto7176200889899739231t_rule @ top_top_set_set_rule @ ord_less_eq_set_rule @ ord_less_eq_set_rule @ F )
=> ( ! [S6: set_rule] :
( ( P @ S6 )
=> ( ( ord_less_eq_set_rule @ ( comple2988633178464755965t_rule @ F ) @ S6 )
=> ( P @ ( F @ S6 ) ) ) )
=> ( ! [M5: set_set_rule] :
( ! [X6: set_rule] :
( ( member_set_rule @ X6 @ M5 )
=> ( P @ X6 ) )
=> ( P @ ( comple5773694076043965236t_rule @ M5 ) ) )
=> ( P @ ( comple2988633178464755965t_rule @ F ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_1246_all__not__in__conv,axiom,
! [A: set_rat] :
( ( ! [X3: rat] :
~ ( member_rat @ X3 @ A ) )
= ( A = bot_bot_set_rat ) ) ).
% all_not_in_conv
thf(fact_1247_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1248_all__not__in__conv,axiom,
! [A: set_rule] :
( ( ! [X3: rule] :
~ ( member_rule @ X3 @ A ) )
= ( A = bot_bot_set_rule ) ) ).
% all_not_in_conv
thf(fact_1249_empty__iff,axiom,
! [C2: rat] :
~ ( member_rat @ C2 @ bot_bot_set_rat ) ).
% empty_iff
thf(fact_1250_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1251_empty__iff,axiom,
! [C2: rule] :
~ ( member_rule @ C2 @ bot_bot_set_rule ) ).
% empty_iff
thf(fact_1252_InterI,axiom,
! [C: set_set_rat,A: rat] :
( ! [X7: set_rat] :
( ( member_set_rat @ X7 @ C )
=> ( member_rat @ A @ X7 ) )
=> ( member_rat @ A @ ( comple4298007329820168263et_rat @ C ) ) ) ).
% InterI
thf(fact_1253_InterI,axiom,
! [C: set_set_nat,A: nat] :
( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ C )
=> ( member_nat @ A @ X7 ) )
=> ( member_nat @ A @ ( comple7806235888213564991et_nat @ C ) ) ) ).
% InterI
thf(fact_1254_InterI,axiom,
! [C: set_set_rule,A: rule] :
( ! [X7: set_rule] :
( ( member_set_rule @ X7 @ C )
=> ( member_rule @ A @ X7 ) )
=> ( member_rule @ A @ ( comple5773694076043965236t_rule @ C ) ) ) ).
% InterI
thf(fact_1255_Inter__iff,axiom,
! [A: rat,C: set_set_rat] :
( ( member_rat @ A @ ( comple4298007329820168263et_rat @ C ) )
= ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ C )
=> ( member_rat @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_1256_Inter__iff,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C )
=> ( member_nat @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_1257_Inter__iff,axiom,
! [A: rule,C: set_set_rule] :
( ( member_rule @ A @ ( comple5773694076043965236t_rule @ C ) )
= ( ! [X3: set_rule] :
( ( member_set_rule @ X3 @ C )
=> ( member_rule @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_1258_image__is__empty,axiom,
! [F: nat > rule,A: set_nat] :
( ( ( image_nat_rule @ F @ A )
= bot_bot_set_rule )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1259_image__is__empty,axiom,
! [F: nat > rat,A: set_nat] :
( ( ( image_nat_rat @ F @ A )
= bot_bot_set_rat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1260_empty__is__image,axiom,
! [F: nat > rule,A: set_nat] :
( ( bot_bot_set_rule
= ( image_nat_rule @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1261_empty__is__image,axiom,
! [F: nat > rat,A: set_nat] :
( ( bot_bot_set_rat
= ( image_nat_rat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1262_image__empty,axiom,
! [F: nat > rule] :
( ( image_nat_rule @ F @ bot_bot_set_nat )
= bot_bot_set_rule ) ).
% image_empty
thf(fact_1263_image__empty,axiom,
! [F: nat > rat] :
( ( image_nat_rat @ F @ bot_bot_set_nat )
= bot_bot_set_rat ) ).
% image_empty
thf(fact_1264_Sup__bot__conv_I1_J,axiom,
! [A: set_set_rat] :
( ( ( comple3890839924845867745et_rat @ A )
= bot_bot_set_rat )
= ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
=> ( X3 = bot_bot_set_rat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1265_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1266_Sup__bot__conv_I1_J,axiom,
! [A: set_set_rule] :
( ( ( comple2146307154184993742t_rule @ A )
= bot_bot_set_rule )
= ( ! [X3: set_rule] :
( ( member_set_rule @ X3 @ A )
=> ( X3 = bot_bot_set_rule ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1267_Sup__bot__conv_I2_J,axiom,
! [A: set_set_rat] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ A ) )
= ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
=> ( X3 = bot_bot_set_rat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1268_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1269_Sup__bot__conv_I2_J,axiom,
! [A: set_set_rule] :
( ( bot_bot_set_rule
= ( comple2146307154184993742t_rule @ A ) )
= ( ! [X3: set_rule] :
( ( member_set_rule @ X3 @ A )
=> ( X3 = bot_bot_set_rule ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1270_atLeast__empty__triv,axiom,
( ( set_or7446828528931440131et_rat @ bot_bot_set_rat )
= top_top_set_set_rat ) ).
% atLeast_empty_triv
thf(fact_1271_atLeast__empty__triv,axiom,
( ( set_or1731685050470061051et_nat @ bot_bot_set_nat )
= top_top_set_set_nat ) ).
% atLeast_empty_triv
thf(fact_1272_atLeast__empty__triv,axiom,
( ( set_or5121885173736058096t_rule @ bot_bot_set_rule )
= top_top_set_set_rule ) ).
% atLeast_empty_triv
thf(fact_1273_boolean__algebra_Ocompl__one,axiom,
( ( uminus613421341184616069et_nat @ top_top_set_set_nat )
= bot_bot_set_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_1274_boolean__algebra_Ocompl__one,axiom,
( ( uminus4253708974368314874t_rule @ top_top_set_set_rule )
= bot_bot_set_set_rule ) ).
% boolean_algebra.compl_one
% Conjectures (1)
thf(conj_0,conjecture,
( ( sset_rule @ rules )
= top_top_set_rule ) ).
%------------------------------------------------------------------------------