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Theorem limelon 4455
Description: A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
Assertion
Ref Expression
limelon  |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  On )

Proof of Theorem limelon
StepHypRef Expression
1 limord 4451 . . 3  |-  ( Lim 
A  ->  Ord  A )
2 elong 4400 . . 3  |-  ( A  e.  B  ->  ( A  e.  On  <->  Ord  A ) )
31, 2syl5ibr 212 . 2  |-  ( A  e.  B  ->  ( Lim  A  ->  A  e.  On ) )
43imp 418 1  |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   Ord word 4391   Oncon0 4392   Lim wlim 4393
This theorem is referenced by:  onzsl  4637  limuni3  4643  tfindsg2  4652  dfom2  4658  rdglim  6439  oalim  6531  omlim  6532  oelim  6533  oalimcl  6558  oaass  6559  omlimcl  6576  odi  6577  omass  6578  oen0  6584  oewordri  6590  oelim2  6593  oelimcl  6598  omabs  6645  r1lim  7444  alephordi  7701  cflm  7876  alephsing  7902  pwcfsdom  8205  winafp  8319  r1limwun  8358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397
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