Theorem limuni2 4375

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 4374	. . 3 |- ( Lim A -> A = U. A )
2		limeq 4355	. . 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 1343   U.cuni 3789   Lim wlim 4342

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-ilim 4347

This theorem is referenced by: (None)