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Theorem ondomen 7754
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 4108 . . . 4  |-  ( x  =  A  ->  ( B  ~<_  x  <->  B  ~<_  A ) )
21rspcev 2960 . . 3  |-  ( ( A  e.  On  /\  B  ~<_  A )  ->  E. x  e.  On  B  ~<_  x )
3 ac10ct 7751 . . 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 7752 . 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 1541    e. wcel 1710   E.wrex 2620   class class class wbr 4104    We wwe 4433   Oncon0 4474   dom cdm 4771    ~<_ cdom 6949   cardccrd 7658
This theorem is referenced by:  numdom  7755  alephnbtwn2  7789  alephsucdom  7796  fictb  7961  cfslb2n  7984  gchaleph2  8388  hargch  8389  inawinalem  8401  rankcf  8489  tskuni  8495  1stcrestlem  17284  2ndcctbss  17287  2ndcomap  17290  2ndcsep  17291  tx1stc  17450  tx2ndc  17451  met2ndci  18170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-riota 6391  df-recs 6475  df-en 6952  df-dom 6953  df-card 7662
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