Theorem limuni2 4369

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

Assertion

Ref	Expression
limuni2	|- ( Lim A -> Lim U. A )

Proof of Theorem limuni2

Step	Hyp	Ref	Expression
1		limuni 4368	. . 3 |- ( Lim A -> A = U. A )
2		limeq 4349	. . 3 |- ( A = U. A -> ( Lim A <-> Lim U. A ) )
3	1, 2	syl 14	. 2 |- ( Lim A -> ( Lim A <-> Lim U. A ) )
4	3	ibi 175	1 |- ( Lim A -> Lim U. A )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1342   U.cuni 3783   Lim wlim 4336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-in 3117  df-ss 3124  df-uni 3784  df-tr 4075  df-iord 4338  df-ilim 4341

This theorem is referenced by: (None)