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Theorem sucidALT 36715
Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 36714 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 4002 . . 3  |-  A  e. 
{ A }
3 elun1 3612 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( { A }  u.  A
) )
42, 3ax-mp 5 . 2  |-  A  e.  ( { A }  u.  A )
5 df-suc 5418 . . 3  |-  suc  A  =  ( A  u.  { A } )
65equncomi 3591 . 2  |-  suc  A  =  ( { A }  u.  A )
74, 6eleqtrri 2491 1  |-  A  e. 
suc  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1844   _Vcvv 3061    u. cun 3414   {csn 3974   suc csuc 5414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-v 3063  df-un 3421  df-in 3423  df-ss 3430  df-sn 3975  df-suc 5418
This theorem is referenced by: (None)
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