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Theorem sucidALT 28984
Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 28983 using translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
sucidALT.1  |-  A  e. 
_V
Assertion
Ref Expression
sucidALT  |-  A  e. 
suc  A

Proof of Theorem sucidALT
StepHypRef Expression
1 sucidALT.1 . . . 4  |-  A  e. 
_V
21snid 3842 . . 3  |-  A  e. 
{ A }
3 elun1 3515 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( { A }  u.  A
) )
42, 3ax-mp 8 . 2  |-  A  e.  ( { A }  u.  A )
5 df-suc 4588 . . 3  |-  suc  A  =  ( A  u.  { A } )
65equncomi 3494 . 2  |-  suc  A  =  ( { A }  u.  A )
74, 6eleqtrri 2510 1  |-  A  e. 
suc  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1726   _Vcvv 2957    u. cun 3319   {csn 3815   suc csuc 4584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-un 3326  df-in 3328  df-ss 3335  df-sn 3821  df-suc 4588
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