Theorem limelon 4950

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 4946	. . 3 |- ( Lim A -> Ord A )
2		elong 4895	. . 3 |- ( A e. B -> ( A e. On <-> Ord A ) )
3	1, 2	syl5ibr 221	. 2 |- ( A e. B -> ( Lim A -> A e. On ) )
4	3	imp 429	1 |- ( ( A e. B /\ Lim A ) -> A e. On )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1819   Ord word 4886   Oncon0 4887   Lim wlim 4888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3478  df-ss 3485  df-uni 4252  df-tr 4551  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892

This theorem is referenced by:  onzsl  6680  limuni3  6686  tfindsg2  6695  dfom2  6701  rdglim  7110  oalim  7200  omlim  7201  oelim  7202  oalimcl  7227  oaass  7228  omlimcl  7245  odi  7246  omass  7247  oen0  7253  oewordri  7259  oelim2  7262  oelimcl  7267  omabs  7314  r1lim  8207  alephordi  8472  cflm  8647  alephsing  8673  pwcfsdom  8975  winafp  9092  r1limwun  9131