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Theorem sucssel 4810
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
StepHypRef Expression
1 sucidg 4796 . 2  |-  ( A  e.  V  ->  A  e.  suc  A )
2 ssel 3349 . 2  |-  ( suc 
A  C_  B  ->  ( A  e.  suc  A  ->  A  e.  B ) )
31, 2syl5com 30 1  |-  ( A  e.  V  ->  ( suc  A  C_  B  ->  A  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756    C_ wss 3327   suc csuc 4720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-v 2973  df-un 3332  df-in 3334  df-ss 3341  df-sn 3877  df-suc 4724
This theorem is referenced by:  suc11  4821  ordelsuc  6430  ordsucelsuc  6432  oaordi  6984  nnaordi  7056  unbnn2  7568  ackbij1b  8407  ackbij2  8411  cflm  8418  isf32lem2  8522  indpi  9075  dfon2lem3  27597
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