Theorem limuni2 6220

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 6219	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2		limeq 6171	. . 3 ⊢ (𝐴 = ∪ 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴))
3	1, 2	syl 17	. 2 ⊢ (Lim 𝐴 → (Lim 𝐴 ↔ Lim ∪ 𝐴))
4	3	ibi 270	1 ⊢ (Lim 𝐴 → Lim ∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  ∪ cuni 4800  Lim wlim 6160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801  df-tr 5137  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-lim 6164

This theorem is referenced by:  rankxplim2  9293  rankxplim3  9294