Theorem limuni2 4444

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 4443	. . 3 |- ( Lim A -> A = U. A )
2		limeq 4424	. . 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 176	1 |- ( Lim A -> Lim U. A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   U.cuni 3850   Lim wlim 4411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4143  df-iord 4413  df-ilim 4416

This theorem is referenced by: (None)