Theorem sucssel 4262

Description: A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)

Assertion

Ref	Expression
sucssel	|- ( A e. V -> ( suc A C_ B -> A e. B ) )

Proof of Theorem sucssel

Step	Hyp	Ref	Expression
1		sucidg 4254	. 2 |- ( A e. V -> A e. suc A )
2		ssel 3022	. 2 |- ( suc A C_ B -> ( A e. suc A -> A e. B ) )
3	1, 2	syl5com 29	1 |- ( A e. V -> ( suc A C_ B -> A e. B ) )

Syntax hints:   -> wi 4   e. wcel 1439   C_ wss 3002   suc csuc 4203

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-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-sn 3458  df-suc 4209

This theorem is referenced by:  ordelsuc  4337  bj-nnelirr  12152