Theorem onuni 4641

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 4640	. 2 |- ( A e. On -> A C_ On )
2		ssonuni 4635	. 2 |- ( A e. On -> ( A C_ On -> U. A e. On ) )
3	1, 2	mpd 13	1 |- ( A e. On -> U. A e. On )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   C_ wss 3220   U.cuni 3935   Oncon0 4508

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-uni 3936  df-tr 4230  df-iord 4511  df-on 4513

This theorem is used by: (None)