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Theorem sucidALT 32751
Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 32750 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 4055 . . 3  |-  A  e. 
{ A }
3 elun1 3671 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( { A }  u.  A
) )
42, 3ax-mp 5 . 2  |-  A  e.  ( { A }  u.  A )
5 df-suc 4884 . . 3  |-  suc  A  =  ( A  u.  { A } )
65equncomi 3650 . 2  |-  suc  A  =  ( { A }  u.  A )
74, 6eleqtrri 2554 1  |-  A  e. 
suc  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767   _Vcvv 3113    u. cun 3474   {csn 4027   suc csuc 4880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481  df-in 3483  df-ss 3490  df-sn 4028  df-suc 4884
This theorem is referenced by: (None)
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