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Theorem eqelsuc 4473
Description: A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
Hypothesis
Ref Expression
eqelsuc.1  |-  A  e. 
_V
Assertion
Ref Expression
eqelsuc  |-  ( A  =  B  ->  A  e.  suc  B )

Proof of Theorem eqelsuc
StepHypRef Expression
1 eqelsuc.1 . . 3  |-  A  e. 
_V
21sucid 4471 . 2  |-  A  e. 
suc  A
3 suceq 4457 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
42, 3syl5eleq 2369 1  |-  ( A  =  B  ->  A  e.  suc  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   suc csuc 4394
This theorem is referenced by:  pssnn  7081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-sn 3646  df-suc 4398
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