Theorem sucssel 4456

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 4448	. 2 |- ( A e. V -> A e. suc A )
2		ssel 3174	. 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 2164   C_ wss 3154   suc csuc 4397

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-suc 4403

This theorem is referenced by:  ordelsuc  4538  sucpw1nss3  7297  3nsssucpw1  7298  bj-nnelirr  15515