Theorem ssonunii 4587

Description: The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.)

Hypothesis

Ref	Expression
ssonuni.1	|- A e. _V

Assertion

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

Proof of Theorem ssonunii

Step	Hyp	Ref	Expression
1		ssonuni.1	. 2 |- A e. _V
2		ssonuni 4586	. 2 |- ( A e. _V -> ( A C_ On -> U. A e. On ) )
3	1, 2	ax-mp 5	1 |- ( A C_ On -> U. A e. On )

Syntax hints:   -> wi 4   e. wcel 2202   _Vcvv 2802   C_ wss 3200   U.cuni 3893   Oncon0 4460

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465

This theorem is referenced by:  bm2.5ii  4594