Theorem onuni 4495

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 4494	. 2 |- ( A e. On -> A C_ On )
2		ssonuni 4489	. 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 2148   C_ wss 3131   U.cuni 3811   Oncon0 4365

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370

This theorem is referenced by: (None)