Theorem ssonuni 4214

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 4213	. 2 |- ( A C_ On -> Ord U. A )
2		uniexg 4175	. . 3 |- ( A e. V -> U. A e. _V )
3		elong 4110	. . 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 145	1 |- ( A e. V -> ( A C_ On -> U. A e. On ) )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   _Vcvv 2557   C_ wss 2917   U.cuni 3580   Ord word 4099   Oncon0 4100

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105

This theorem is referenced by:  ssonunii  4215  onun2  4216  onuni  4220  iunon  5899