Theorem limelon 6406

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 6402	. . 3 ⊢ (Lim 𝐴 → Ord 𝐴)
2		elong 6349	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	imbitrrid 248	. 2 ⊢ (𝐴 ∈ 𝐵 → (Lim 𝐴 → 𝐴 ∈ On))
4	3	imp 410	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141  Ord word 6340  Oncon0 6341  Lim wlim 6342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-v 3455  df-ss 3919  df-uni 4863  df-tr 5205  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-ord 6344  df-on 6345  df-lim 6346

This theorem is referenced by:  onzsl  7821  limuni3  7827  tfindsg2  7837  dfom2  7843  rdglim  8391  oalim  8495  omlim  8496  oelim  8497  oalimcl  8523  oaass  8524  omlimcl  8541  odi  8542  omass  8543  oen0  8550  oewordri  8556  oelim2  8559  oelimcl  8564  omabs  8615  r1lim  9724  alephordi  10024  cflm  10200  alephsing  10227  pwcfsdom  10535  winafp  10649  r1limwun  10688  omlimcl2  43780  oeord2lim  43847