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Theorem limelon 4647
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 4643 . . 3  |-  ( Lim 
A  ->  Ord  A )
2 elong 4592 . . 3  |-  ( A  e.  B  ->  ( A  e.  On  <->  Ord  A ) )
31, 2syl5ibr 214 . 2  |-  ( A  e.  B  ->  ( Lim  A  ->  A  e.  On ) )
43imp 420 1  |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   Ord word 4583   Oncon0 4584   Lim wlim 4585
This theorem is referenced by:  onzsl  4829  limuni3  4835  tfindsg2  4844  dfom2  4850  rdglim  6687  oalim  6779  omlim  6780  oelim  6781  oalimcl  6806  oaass  6807  omlimcl  6824  odi  6825  omass  6826  oen0  6832  oewordri  6838  oelim2  6841  oelimcl  6846  omabs  6893  r1lim  7701  alephordi  7960  cflm  8135  alephsing  8161  pwcfsdom  8463  winafp  8577  r1limwun  8616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-in 3329  df-ss 3336  df-uni 4018  df-tr 4306  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589
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