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Theorem sucidVD 28964
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 4487 is sucidVD 28964 without virtual deductions and was automatically derived from sucidVD 28964.
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 3680 . . 3  |-  A  e. 
{ A }
3 elun2 3356 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( A  u.  { A }
) )
42, 3e0_ 28861 . 2  |-  A  e.  ( A  u.  { A } )
5 df-suc 4414 . 2  |-  suc  A  =  ( A  u.  { A } )
64, 5eleqtrri 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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