Theorem onuni 4585

Description: The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)

Assertion

Ref	Expression
onuni	|- ( A e. On -> U. A e. On )

Proof of Theorem onuni

Step	Hyp	Ref	Expression
1		onss 4584	. 2 |- ( A e. On -> A C_ On )
2		ssonuni 4579	. 2 |- ( A e. On -> ( A C_ On -> U. A e. On ) )
3	1, 2	mpd 13	1 |- ( A e. On -> U. A e. On )

Syntax hints:   -> wi 4   e. wcel 2200   C_ wss 3197   U.cuni 3887   Oncon0 4453

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-uni 3888  df-tr 4182  df-iord 4456  df-on 4458

This theorem is referenced by: (None)