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Theorem sucssel 4262
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 4254 . 2  |-  ( A  e.  V  ->  A  e.  suc  A )
2 ssel 3022 . 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 1439    C_ wss 3002   suc csuc 4203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-sn 3458  df-suc 4209
This theorem is referenced by:  ordelsuc  4337  bj-nnelirr  12152
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