Theorem limelon 5448

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

Step	Hyp	Ref	Expression
1		limord 5444	. . 3 |- ( Lim A -> Ord A )
2		elong 5393	. . 3 |- ( A e. B -> ( A e. On <-> Ord A ) )
3	1, 2	syl5ibr 224	. 2 |- ( A e. B -> ( Lim A -> A e. On ) )
4	3	imp 430	1 |- ( ( A e. B /\ Lim A ) -> A e. On )

Syntax hints:   -> wi 4   /\ wa 370   e. wcel 1872   Ord word 5384   Oncon0 5385   Lim wlim 5386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408

This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-v 3024  df-in 3386  df-ss 3393  df-uni 4163  df-tr 4462  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-ord 5388  df-on 5389  df-lim 5390

This theorem is referenced by:  onzsl  6631  limuni3  6637  tfindsg2  6646  dfom2  6652  rdglim  7099  oalim  7189  omlim  7190  oelim  7191  oalimcl  7216  oaass  7217  omlimcl  7234  odi  7235  omass  7236  oen0  7242  oewordri  7248  oelim2  7251  oelimcl  7256  omabs  7303  r1lim  8195  alephordi  8456  cflm  8631  alephsing  8657  pwcfsdom  8959  winafp  9073  r1limwun  9112