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Theorem sucidALT 28647
Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 28646 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 3667 . . 3  |-  A  e. 
{ A }
3 elun1 3342 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( { A }  u.  A
) )
42, 3ax-mp 8 . 2  |-  A  e.  ( { A }  u.  A )
5 df-suc 4398 . . 3  |-  suc  A  =  ( A  u.  { A } )
65equncomi 3321 . 2  |-  suc  A  =  ( { A }  u.  A )
74, 6eleqtrri 2356 1  |-  A  e. 
suc  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   _Vcvv 2788    u. cun 3150   {csn 3640   suc csuc 4394
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-in 3159  df-ss 3166  df-sn 3646  df-suc 4398
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