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Theorem sucssel 4436
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 4428 . 2  |-  ( A  e.  V  ->  A  e.  suc  A )
2 ssel 3161 . 2  |-  ( suc 
A  C_  B  ->  ( A  e.  suc  A  ->  A  e.  B ) )
31, 2syl5com 29 1  |-  ( A  e.  V  ->  ( suc  A  C_  B  ->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2158    C_ wss 3141   suc csuc 4377
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-suc 4383
This theorem is referenced by:  ordelsuc  4516  sucpw1nss3  7247  3nsssucpw1  7248  bj-nnelirr  14945
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