Theorem limelon 6450

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 6446	. . 3 ⊢ (Lim 𝐴 → Ord 𝐴)
2		elong 6394	. . 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 2106  Ord word 6385  Oncon0 6386  Lim wlim 6387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-ss 3980  df-uni 4913  df-tr 5266  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391

This theorem is referenced by:  onzsl  7867  limuni3  7873  tfindsg2  7883  dfom2  7889  rdglim  8465  oalim  8569  omlim  8570  oelim  8571  oalimcl  8597  oaass  8598  omlimcl  8615  odi  8616  omass  8617  oen0  8623  oewordri  8629  oelim2  8632  oelimcl  8637  omabs  8688  r1lim  9810  alephordi  10112  cflm  10288  alephsing  10314  pwcfsdom  10621  winafp  10735  r1limwun  10774  omlimcl2  43231  oeord2lim  43299