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Theorem eqelsuc 5490
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 5488 . 2  |-  A  e. 
suc  A
3 suceq 5474 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
42, 3syl5eleq 2496 1  |-  ( A  =  B  ->  A  e.  suc  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   _Vcvv 3058   suc csuc 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-un 3418  df-sn 3972  df-suc 5415
This theorem is referenced by:  pssnn  7772
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