Theorem limuni2 4432

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 4431	. . 3 |- ( Lim A -> A = U. A )
2		limeq 4412	. . 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 1364   U.cuni 3839   Lim wlim 4399

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-in 3163  df-ss 3170  df-uni 3840  df-tr 4132  df-iord 4401  df-ilim 4404

This theorem is referenced by: (None)