Theorem sucssel 5819

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 5803	. 2 |- ( A e. V -> A e. suc A )
2		ssel 3597	. 2 |- ( suc A C_ B -> ( A e. suc A -> A e. B ) )
3	1, 2	syl5com 31	1 |- ( A e. V -> ( suc A C_ B -> A e. B ) )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-suc 5729

This theorem is referenced by:  suc11  5831  ordelsuc  7020  ordsucelsuc  7022  oaordi  7626  nnaordi  7698  unbnn2  8217  ackbij1b  9061  ackbij2  9065  cflm  9072  isf32lem2  9176  indpi  9729  dfon2lem3  31690