Theorem limord 4186

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 4161	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
2	1	simp1bi 954	1 ⊢ (Lim 𝐴 → Ord 𝐴)

Syntax hints:   → wi 4   = wceq 1285   ∈ wcel 1434  ∅c0 3269  ∪ cuni 3627  Ord word 4153  Lim wlim 4155

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104

This theorem depends on definitions:  df-bi 115  df-3an 922  df-ilim 4160

This theorem is referenced by:  limelon  4190  nlimsucg  4345