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Theorem elharval 7989
Description: The Hartogs number of a set is greater than all ordinals which inject into it. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
elharval  |-  ( Y  e.  (har `  X
)  <->  ( Y  e.  On  /\  Y  ~<_  X ) )

Proof of Theorem elharval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elfvex 5893 . 2  |-  ( Y  e.  (har `  X
)  ->  X  e.  _V )
2 reldom 7522 . . . 4  |-  Rel  ~<_
32brrelex2i 5041 . . 3  |-  ( Y  ~<_  X  ->  X  e.  _V )
43adantl 466 . 2  |-  ( ( Y  e.  On  /\  Y  ~<_  X )  ->  X  e.  _V )
5 harval 7988 . . . 4  |-  ( X  e.  _V  ->  (har `  X )  =  {
y  e.  On  | 
y  ~<_  X } )
65eleq2d 2537 . . 3  |-  ( X  e.  _V  ->  ( Y  e.  (har `  X
)  <->  Y  e.  { y  e.  On  |  y  ~<_  X } ) )
7 breq1 4450 . . . 4  |-  ( y  =  Y  ->  (
y  ~<_  X  <->  Y  ~<_  X ) )
87elrab 3261 . . 3  |-  ( Y  e.  { y  e.  On  |  y  ~<_  X }  <->  ( Y  e.  On  /\  Y  ~<_  X ) )
96, 8syl6bb 261 . 2  |-  ( X  e.  _V  ->  ( Y  e.  (har `  X
)  <->  ( Y  e.  On  /\  Y  ~<_  X ) ) )
101, 4, 9pm5.21nii 353 1  |-  ( Y  e.  (har `  X
)  <->  ( Y  e.  On  /\  Y  ~<_  X ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1767   {crab 2818   _Vcvv 3113   class class class wbr 4447   Oncon0 4878   ` cfv 5588    ~<_ cdom 7514  harchar 7982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-recs 7042  df-en 7517  df-dom 7518  df-oi 7935  df-har 7984
This theorem is referenced by:  harndom  7990  harcard  8359  cardprclem  8360  cardaleph  8470  dfac12lem2  8524  hsmexlem1  8806  pwcfsdom  8958  pwfseqlem5  9041  hargch  9051  harinf  30608  harn0  30683
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