Theorem ssonunii 4321

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 4320	. 2 |- ( A e. _V -> ( A C_ On -> U. A e. On ) )
3	1, 2	ax-mp 7	1 |- ( A C_ On -> U. A e. On )

Syntax hints:   -> wi 4   e. wcel 1439   _Vcvv 2622   C_ wss 3002   U.cuni 3661   Oncon0 4201

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-un 4271

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-in 3008  df-ss 3015  df-uni 3662  df-tr 3945  df-iord 4204  df-on 4206

This theorem is referenced by:  bm2.5ii  4328