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Theorem elharval 8078
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 5908 . 2  |-  ( Y  e.  (har `  X
)  ->  X  e.  _V )
2 reldom 7583 . . . 4  |-  Rel  ~<_
32brrelex2i 4896 . . 3  |-  ( Y  ~<_  X  ->  X  e.  _V )
43adantl 467 . 2  |-  ( ( Y  e.  On  /\  Y  ~<_  X )  ->  X  e.  _V )
5 harval 8077 . . . 4  |-  ( X  e.  _V  ->  (har `  X )  =  {
y  e.  On  | 
y  ~<_  X } )
65eleq2d 2499 . . 3  |-  ( X  e.  _V  ->  ( Y  e.  (har `  X
)  <->  Y  e.  { y  e.  On  |  y  ~<_  X } ) )
7 breq1 4429 . . . 4  |-  ( y  =  Y  ->  (
y  ~<_  X  <->  Y  ~<_  X ) )
87elrab 3235 . . 3  |-  ( Y  e.  { y  e.  On  |  y  ~<_  X }  <->  ( Y  e.  On  /\  Y  ~<_  X ) )
96, 8syl6bb 264 . 2  |-  ( X  e.  _V  ->  ( Y  e.  (har `  X
)  <->  ( Y  e.  On  /\  Y  ~<_  X ) ) )
101, 4, 9pm5.21nii 354 1  |-  ( Y  e.  (har `  X
)  <->  ( Y  e.  On  /\  Y  ~<_  X ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    e. wcel 1870   {crab 2786   _Vcvv 3087   class class class wbr 4426   Oncon0 5442   ` cfv 5601    ~<_ cdom 7575  harchar 8071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-wrecs 7036  df-recs 7098  df-en 7578  df-dom 7579  df-oi 8025  df-har 8073
This theorem is referenced by:  harndom  8079  harcard  8411  cardprclem  8412  cardaleph  8518  dfac12lem2  8572  hsmexlem1  8854  pwcfsdom  9006  pwfseqlem5  9087  hargch  9097  harinf  35594  harn0  35666
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