Theorem limelon 6454

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 6450	. . 3 ⊢ (Lim 𝐴 → Ord 𝐴)
2		elong 6396	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	imbitrrid 249	. 2 ⊢ (𝐴 ∈ 𝐵 → (Lim 𝐴 → 𝐴 ∈ On))
4	3	imp 412	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  Ord word 6387  Oncon0 6388  Lim wlim 6389

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-ss 3901  df-uni 4873  df-tr 5219  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392  df-lim 6393

This theorem is used by:  onzsl  7891  limuni3  7897  tfindsg2  7907  dfom2  7913  rdglim  8472  oalim  8578  omlim  8579  oelim  8580  oalimcl  8606  oaass  8607  omlimcl  8624  odi  8625  omass  8626  oen0  8633  oewordri  8639  oelim2  8642  oelimcl  8647  omabs  8698  r1lim  9816  alephordi  10192  cflm  10366  alephsing  10393  pwcfsdom  10707  winafp  10821  r1limwun  10860  omlimcl2  44505  oeord2lim  44572