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Theorem isnumi 7826
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 4208 . . 3  |-  ( x  =  A  ->  (
x  ~~  B  <->  A  ~~  B ) )
21rspcev 3045 . 2  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  E. x  e.  On  x  ~~  B )
3 isnum2 7825 . 2  |-  ( B  e.  dom  card  <->  E. x  e.  On  x  ~~  B
)
42, 3sylibr 204 1  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  dom  card )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   E.wrex 2699   class class class wbr 4205   Oncon0 4574   dom cdm 4871    ~~ cen 7099   cardccrd 7815
This theorem is referenced by:  finnum  7828  onenon  7829  tskwe  7830  xpnum  7831  isnum3  7834  dfac8alem  7903  cdanum  8072  fin67  8268  isfin7-2  8269  gchacg  8540  gch2  8547  znnen  12805  qnnen  12806  met1stc  18544  re2ndc  18825  uniiccdif  19463  dyadmbl  19485  opnmblALT  19488  mbfimaopnlem  19540  aannenlem3  20240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-fun 5449  df-fn 5450  df-f 5451  df-en 7103  df-card 7819
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