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Theorem isnumi 8318
Description: A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
isnumi  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  dom  card )

Proof of Theorem isnumi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 breq1 4442 . . 3  |-  ( x  =  A  ->  (
x  ~~  B  <->  A  ~~  B ) )
21rspcev 3207 . 2  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  E. x  e.  On  x  ~~  B )
3 isnum2 8317 . 2  |-  ( B  e.  dom  card  <->  E. x  e.  On  x  ~~  B
)
42, 3sylibr 212 1  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  dom  card )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   E.wrex 2805   class class class wbr 4439   Oncon0 4867   dom cdm 4988    ~~ cen 7506   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-en 7510  df-card 8311
This theorem is referenced by:  finnum  8320  onenon  8321  tskwe  8322  xpnum  8323  isnum3  8326  dfac8alem  8401  cdanum  8570  fin67  8766  isfin7-2  8767  gch2  9042  gchacg  9047  znnen  14030  qnnen  14031  met1stc  21190  re2ndc  21472  uniiccdif  22153  dyadmbl  22175  opnmblALT  22178  mbfimaopnlem  22228  aannenlem3  22892
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