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Theorem isnumi 7766
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 4156 . . 3  |-  ( x  =  A  ->  (
x  ~~  B  <->  A  ~~  B ) )
21rspcev 2995 . 2  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  E. x  e.  On  x  ~~  B )
3 isnum2 7765 . 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 1717   E.wrex 2650   class class class wbr 4153   Oncon0 4522   dom cdm 4818    ~~ cen 7042   cardccrd 7755
This theorem is referenced by:  finnum  7768  onenon  7769  tskwe  7770  xpnum  7771  isnum3  7774  dfac8alem  7843  cdanum  8012  fin67  8208  isfin7-2  8209  gchacg  8480  gch2  8487  znnen  12739  qnnen  12740  met1stc  18441  re2ndc  18703  uniiccdif  19337  dyadmbl  19359  opnmblALT  19362  mbfimaopnlem  19414  aannenlem3  20114
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-fun 5396  df-fn 5397  df-f 5398  df-en 7046  df-card 7759
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