Theorem onuni 4507

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 4506	. 2 |- ( A e. On -> A C_ On )
2		ssonuni 4501	. 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 2159   C_ wss 3143   U.cuni 3823   Oncon0 4377

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-un 4447

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-in 3149  df-ss 3156  df-uni 3824  df-tr 4116  df-iord 4380  df-on 4382

This theorem is referenced by: (None)