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Theorem ondomen 7878
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 4180 . . . 4  |-  ( x  =  A  ->  ( B  ~<_  x  <->  B  ~<_  A ) )
21rspcev 3016 . . 3  |-  ( ( A  e.  On  /\  B  ~<_  A )  ->  E. x  e.  On  B  ~<_  x )
3 ac10ct 7875 . . 3  |-  ( E. x  e.  On  B  ~<_  x  ->  E. r  r  We  B )
42, 3syl 16 . 2  |-  ( ( A  e.  On  /\  B  ~<_  A )  ->  E. r  r  We  B )
5 ween 7876 . 2  |-  ( B  e.  dom  card  <->  E. r 
r  We  B )
64, 5sylibr 204 1  |-  ( ( A  e.  On  /\  B  ~<_  A )  ->  B  e.  dom  card )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    e. wcel 1721   E.wrex 2671   class class class wbr 4176    We wwe 4504   Oncon0 4545   dom cdm 4841    ~<_ cdom 7070   cardccrd 7782
This theorem is referenced by:  numdom  7879  alephnbtwn2  7913  alephsucdom  7920  fictb  8085  cfslb2n  8108  gchaleph2  8511  hargch  8512  inawinalem  8524  rankcf  8612  tskuni  8618  1stcrestlem  17472  2ndcctbss  17475  2ndcomap  17478  2ndcsep  17479  tx1stc  17639  tx2ndc  17640  met2ndci  18509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6512  df-recs 6596  df-en 7073  df-dom 7074  df-card 7786
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