Theorem limuni2 6447

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 6446	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2		limeq 6395	. . 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∪ cuni 4870  Lim wlim 6384

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-v 3456  df-ss 3901  df-uni 4871  df-tr 5217  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-ord 6386  df-lim 6388

This theorem is used by:  rankxplim2  9927  rankxplim3  9928