Theorem limuni 4431

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

Assertion

Ref	Expression
limuni	⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)

Proof of Theorem limuni

Step	Hyp	Ref	Expression
1		dflim2 4405	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
2	1	simp3bi 1016	1 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  ∅c0 3450  ∪ cuni 3839  Ord word 4397  Lim wlim 4399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 982  df-ilim 4404

This theorem is referenced by:  limuni2  4432  nlimsucg  4602  freccllem  6460  frecfcllem  6462  frecsuclem  6464