Theorem limelon 6382

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 6378	. . 3 ⊢ (Lim 𝐴 → Ord 𝐴)
2		elong 6325	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	imbitrrid 247	. 2 ⊢ (𝐴 ∈ 𝐵 → (Lim 𝐴 → 𝐴 ∈ On))
4	3	imp 407	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  Ord word 6316  Oncon0 6317  Lim wlim 6318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-v 3434  df-ss 3907  df-uni 4846  df-tr 5187  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6320  df-on 6321  df-lim 6322

This theorem is referenced by:  onzsl  7793  limuni3  7799  tfindsg2  7809  dfom2  7815  rdglim  8362  oalim  8464  omlim  8465  oelim  8466  oalimcl  8492  oaass  8493  omlimcl  8510  odi  8511  omass  8512  oen0  8519  oewordri  8525  oelim2  8528  oelimcl  8533  omabs  8584  r1lim  9694  alephordi  9994  cflm  10170  alephsing  10196  pwcfsdom  10504  winafp  10618  r1limwun  10657  omlimcl2  43694  oeord2lim  43761