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Theorem isnumi 8330
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 4440 . . 3  |-  ( x  =  A  ->  (
x  ~~  B  <->  A  ~~  B ) )
21rspcev 3196 . 2  |-  ( ( A  e.  On  /\  A  ~~  B )  ->  E. x  e.  On  x  ~~  B )
3 isnum2 8329 . 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 369    e. wcel 1804   E.wrex 2794   class class class wbr 4437   Oncon0 4868   dom cdm 4989    ~~ cen 7515   cardccrd 8319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-fun 5580  df-fn 5581  df-f 5582  df-en 7519  df-card 8323
This theorem is referenced by:  finnum  8332  onenon  8333  tskwe  8334  xpnum  8335  isnum3  8338  dfac8alem  8413  cdanum  8582  fin67  8778  isfin7-2  8779  gch2  9056  gchacg  9061  znnen  13823  qnnen  13824  met1stc  20897  re2ndc  21179  uniiccdif  21860  dyadmbl  21882  opnmblALT  21885  mbfimaopnlem  21935  aannenlem3  22598
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