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Theorem sucidALT 28963
Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 28962 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 3680 . . 3  |-  A  e. 
{ A }
3 elun1 3355 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( { A }  u.  A
) )
42, 3ax-mp 8 . 2  |-  A  e.  ( { A }  u.  A )
5 df-suc 4414 . . 3  |-  suc  A  =  ( A  u.  { A } )
65equncomi 3334 . 2  |-  suc  A  =  ( { A }  u.  A )
74, 6eleqtrri 2369 1  |-  A  e. 
suc  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   _Vcvv 2801    u. cun 3163   {csn 3653   suc csuc 4410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-sn 3659  df-suc 4414
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