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Theorem eqelsuc 4954
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 4952 . 2  |-  A  e. 
suc  A
3 suceq 4938 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
42, 3syl5eleq 2556 1  |-  ( A  =  B  ->  A  e.  suc  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3108   suc csuc 4875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-v 3110  df-un 3476  df-sn 4023  df-suc 4879
This theorem is referenced by:  pssnn  7730
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