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Theorem sucssel 4501
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 4486 . 2  |-  ( A  e.  V  ->  A  e.  suc  A )
2 ssel 3187 . 2  |-  ( suc 
A  C_  B  ->  ( A  e.  suc  A  ->  A  e.  B ) )
31, 2syl5com 26 1  |-  ( A  e.  V  ->  ( suc  A  C_  B  ->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    C_ wss 3165   suc csuc 4410
This theorem is referenced by:  suc11  4512  ordelsuc  4627  ordsucelsuc  4629  oaordi  6560  nnaordi  6632  unbnn2  7130  ackbij1b  7881  ackbij2  7885  cflm  7892  isf32lem2  7996  indpi  8547  dfon2lem3  24212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-sn 3659  df-suc 4414
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