Theorem limelon 6388

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 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)

Proof of Theorem limelon

Step	Hyp	Ref	Expression
1		limord 6384	. . 3 ⊢ (Lim 𝐴 → Ord 𝐴)
2		elong 6331	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	imbitrrid 246	. 2 ⊢ (𝐴 ∈ 𝐵 → (Lim 𝐴 → 𝐴 ∈ On))
4	3	imp 406	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Ord word 6322  Oncon0 6323  Lim wlim 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3431  df-ss 3906  df-uni 4851  df-tr 5193  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327  df-lim 6328

This theorem is referenced by:  onzsl  7797  limuni3  7803  tfindsg2  7813  dfom2  7819  rdglim  8365  oalim  8467  omlim  8468  oelim  8469  oalimcl  8495  oaass  8496  omlimcl  8513  odi  8514  omass  8515  oen0  8522  oewordri  8528  oelim2  8531  oelimcl  8536  omabs  8587  r1lim  9696  alephordi  9996  cflm  10172  alephsing  10198  pwcfsdom  10506  winafp  10620  r1limwun  10659  omlimcl2  43670  oeord2lim  43737