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Theorem eqelsuc 5523
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 5521 . 2  |-  A  e. 
suc  A
3 suceq 5507 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
42, 3syl5eleq 2513 1  |-  ( A  =  B  ->  A  e.  suc  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872   _Vcvv 3080   suc csuc 5444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-un 3441  df-sn 3999  df-suc 5448
This theorem is referenced by:  pssnn  7799
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