Theorem sucssel 4459

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 4451	. 2 |- ( A e. V -> A e. suc A )
2		ssel 3177	. 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 2167   C_ wss 3157   suc csuc 4400

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-suc 4406

This theorem is referenced by:  ordelsuc  4541  sucpw1nss3  7302  3nsssucpw1  7303  bj-nnelirr  15599