Theorem ssonuni 4536

Description: The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)

Assertion

Ref	Expression
ssonuni	|- ( A e. V -> ( A C_ On -> U. A e. On ) )

Proof of Theorem ssonuni

Step	Hyp	Ref	Expression
1		ssorduni 4535	. 2 |- ( A C_ On -> Ord U. A )
2		uniexg 4486	. . 3 |- ( A e. V -> U. A e. _V )
3		elong 4420	. . 3 |- ( U. A e. _V -> ( U. A e. On <-> Ord U. A ) )
4	2, 3	syl 14	. 2 |- ( A e. V -> ( U. A e. On <-> Ord U. A ) )
5	1, 4	imbitrrid 156	1 |- ( A e. V -> ( A C_ On -> U. A e. On ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2176   _Vcvv 2772   C_ wss 3166   U.cuni 3850   Ord word 4409   Oncon0 4410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415

This theorem is referenced by:  ssonunii  4537  onun2  4538  onuni  4542  iunon  6370