Theorem sucssel 4402

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 4394	. 2 |- ( A e. V -> A e. suc A )
2		ssel 3136	. 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 2136   C_ wss 3116   suc csuc 4343

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-suc 4349

This theorem is referenced by:  ordelsuc  4482  sucpw1nss3  7191  3nsssucpw1  7192  bj-nnelirr  13835