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Theorem sucssel 4976
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 4962 . 2  |-  ( A  e.  V  ->  A  e.  suc  A )
2 ssel 3503 . 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 1767    C_ wss 3481   suc csuc 4886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-un 3486  df-in 3488  df-ss 3495  df-sn 4034  df-suc 4890
This theorem is referenced by:  suc11  4987  ordelsuc  6650  ordsucelsuc  6652  oaordi  7207  nnaordi  7279  unbnn2  7789  ackbij1b  8631  ackbij2  8635  cflm  8642  isf32lem2  8746  indpi  9297  dfon2lem3  29135
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