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Theorem sucidVD 28648
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 4471 is sucidVD 28648 without virtual deductions and was automatically derived from sucidVD 28648.
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 3667 . . 3  |-  A  e. 
{ A }
3 elun2 3343 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( A  u.  { A }
) )
42, 3e0_ 28547 . 2  |-  A  e.  ( A  u.  { A } )
5 df-suc 4398 . 2  |-  suc  A  =  ( A  u.  { A } )
64, 5eleqtrri 2356 1  |-  A  e. 
suc  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   _Vcvv 2788    u. cun 3150   {csn 3640   suc csuc 4394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-sn 3646  df-suc 4398
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