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Theorem limelon 4534
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 4530 . . 3  |-  ( Lim 
A  ->  Ord  A )
2 elong 4479 . . 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 1710   Ord word 4470   Oncon0 4471   Lim wlim 4472
This theorem is referenced by:  onzsl  4716  limuni3  4722  tfindsg2  4731  dfom2  4737  rdglim  6523  oalim  6615  omlim  6616  oelim  6617  oalimcl  6642  oaass  6643  omlimcl  6660  odi  6661  omass  6662  oen0  6668  oewordri  6674  oelim2  6677  oelimcl  6682  omabs  6729  r1lim  7531  alephordi  7788  cflm  7963  alephsing  7989  pwcfsdom  8292  winafp  8406  r1limwun  8445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-v 2866  df-in 3235  df-ss 3242  df-uni 3907  df-tr 4193  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476
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