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Theorem sucidVD 28984
Description: A set belongs to its successor. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sucid 4660 is sucidVD 28984 without virtual deductions and was automatically derived from sucidVD 28984.
h1::  |-  A  e.  _V
2:1:  |-  A  e.  { A }
3:2:  |-  A  e.  ( A  u.  { A } )
4::  |-  suc  A  =  ( A  u.  { A } )
qed:3,4:  |-  A  e.  suc  A
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
sucidVD.1  |-  A  e. 
_V
Assertion
Ref Expression
sucidVD  |-  A  e. 
suc  A

Proof of Theorem sucidVD
StepHypRef Expression
1 sucidVD.1 . . . 4  |-  A  e. 
_V
21snid 3841 . . 3  |-  A  e. 
{ A }
3 elun2 3515 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( A  u.  { A }
) )
42, 3e0_ 28884 . 2  |-  A  e.  ( A  u.  { A } )
5 df-suc 4587 . 2  |-  suc  A  =  ( A  u.  { A } )
64, 5eleqtrri 2509 1  |-  A  e. 
suc  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2956    u. cun 3318   {csn 3814   suc csuc 4583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-in 3327  df-ss 3334  df-sn 3820  df-suc 4587
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