Theorem limuni2 6426

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 6425	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2		limeq 6374	. . 3 ⊢ (𝐴 = ∪ 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴))
3	1, 2	syl 18	. 2 ⊢ (Lim 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴))
4	3	ibi 270	1 ⊢ (Lim 𝐴 → Lim ∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570  ∪ cuni 4873  Lim wlim 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-v 3457  df-ss 3923  df-uni 4874  df-tr 5220  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-lim 6367

This theorem is referenced by:  rankxplim2  9853  rankxplim3  9854