Theorem limuni2 6413

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 6412	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2		limeq 6362	. . 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 1563  ∪ cuni 4868  Lim wlim 6351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-v 3459  df-ss 3924  df-uni 4869  df-tr 5213  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-ord 6353  df-lim 6355

This theorem is referenced by:  rankxplim2  9840  rankxplim3  9841