Theorem limord 4442

Description: A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)

Assertion

Ref	Expression
limord	⊢ (Lim 𝐴 → Ord 𝐴)

Proof of Theorem limord

Step	Hyp	Ref	Expression
1		dflim2 4417	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
2	1	simp1bi 1015	1 ⊢ (Lim 𝐴 → Ord 𝐴)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2176  ∅c0 3460  ∪ cuni 3850  Ord word 4409  Lim wlim 4411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117  df-3an 983  df-ilim 4416

This theorem is referenced by:  limelon  4446  nlimsucg  4614