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Theorem sucssel 4960
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 4946 . 2  |-  ( A  e.  V  ->  A  e.  suc  A )
2 ssel 3483 . 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 1804    C_ wss 3461   suc csuc 4870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-un 3466  df-in 3468  df-ss 3475  df-sn 4015  df-suc 4874
This theorem is referenced by:  suc11  4971  ordelsuc  6640  ordsucelsuc  6642  oaordi  7197  nnaordi  7269  unbnn2  7779  ackbij1b  8622  ackbij2  8626  cflm  8633  isf32lem2  8737  indpi  9288  dfon2lem3  29193
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