Theorem limuni2 6433

Description: The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)

Assertion

Ref	Expression
limuni2	⊢ (Lim 𝐴 → Lim ∪ 𝐴)

Proof of Theorem limuni2

Step	Hyp	Ref	Expression
1		limuni 6432	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2		limeq 6383	. . 3 ⊢ (𝐴 = ∪ 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴))
3	1, 2	syl 17	. 2 ⊢ (Lim 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴))
4	3	ibi 266	1 ⊢ (Lim 𝐴 → Lim ∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533  ∪ cuni 4909  Lim wlim 6372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-v 3463  df-ss 3961  df-uni 4910  df-tr 5267  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374  df-lim 6376

This theorem is referenced by:  rankxplim2  9905  rankxplim3  9906