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Theorem ondomen 7664
Description: If a set is dominated by an ordinal, then it is numerable. (Contributed by Mario Carneiro, 5-Jan-2013.)
Assertion
Ref Expression
ondomen  |-  ( ( A  e.  On  /\  B  ~<_  A )  ->  B  e.  dom  card )

Proof of Theorem ondomen
Dummy variables  x  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4027 . . . 4  |-  ( x  =  A  ->  ( B  ~<_  x  <->  B  ~<_  A ) )
21rspcev 2884 . . 3  |-  ( ( A  e.  On  /\  B  ~<_  A )  ->  E. x  e.  On  B  ~<_  x )
3 ac10ct 7661 . . 3  |-  ( E. x  e.  On  B  ~<_  x  ->  E. r  r  We  B )
42, 3syl 15 . 2  |-  ( ( A  e.  On  /\  B  ~<_  A )  ->  E. r  r  We  B )
5 ween 7662 . 2  |-  ( B  e.  dom  card  <->  E. r 
r  We  B )
64, 5sylibr 203 1  |-  ( ( A  e.  On  /\  B  ~<_  A )  ->  B  e.  dom  card )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    e. wcel 1684   E.wrex 2544   class class class wbr 4023    We wwe 4351   Oncon0 4392   dom cdm 4689    ~<_ cdom 6861   cardccrd 7568
This theorem is referenced by:  numdom  7665  alephnbtwn2  7699  alephsucdom  7706  fictb  7871  cfslb2n  7894  gchaleph2  8298  hargch  8299  inawinalem  8311  rankcf  8399  tskuni  8405  1stcrestlem  17178  2ndcctbss  17181  2ndcomap  17184  2ndcsep  17185  tx1stc  17344  tx2ndc  17345  met2ndci  18068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-en 6864  df-dom 6865  df-card 7572
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