Theorem ssonuni 4472

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 4471	. 2 |- ( A C_ On -> Ord U. A )
2		uniexg 4424	. . 3 |- ( A e. V -> U. A e. _V )
3		elong 4358	. . 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	syl5ibr 155	1 |- ( A e. V -> ( A C_ On -> U. A e. On ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2141   _Vcvv 2730   C_ wss 3121   U.cuni 3796   Ord word 4347   Oncon0 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by:  ssonunii  4473  onun2  4474  onuni  4478  iunon  6263