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Theorem ondomen 8407
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 4444 . . . 4  |-  ( x  =  A  ->  ( B  ~<_  x  <->  B  ~<_  A ) )
21rspcev 3207 . . 3  |-  ( ( A  e.  On  /\  B  ~<_  A )  ->  E. x  e.  On  B  ~<_  x )
3 ac10ct 8404 . . 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 8405 . 2  |-  ( B  e.  dom  card  <->  E. r 
r  We  B )
64, 5sylibr 212 1  |-  ( ( A  e.  On  /\  B  ~<_  A )  ->  B  e.  dom  card )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1591    e. wcel 1762   E.wrex 2808   class class class wbr 4440    We wwe 4830   Oncon0 4871   dom cdm 4992    ~<_ cdom 7504   cardccrd 8305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-recs 7032  df-en 7507  df-dom 7508  df-card 8309
This theorem is referenced by:  numdom  8408  alephnbtwn2  8442  alephsucdom  8449  fictb  8614  cfslb2n  8637  gchaleph2  9039  hargch  9040  inawinalem  9056  rankcf  9144  tskuni  9150  1stcrestlem  19712  2ndcctbss  19715  2ndcomap  19718  2ndcsep  19719  tx1stc  19879  tx2ndc  19880  met2ndci  20753
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