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Theorem sucidVD 37130
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 L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sucid 5518 is sucidVD 37130 without virtual deductions and was automatically derived from sucidVD 37130.
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 4024 . . 3  |-  A  e. 
{ A }
3 elun2 3634 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( A  u.  { A }
) )
42, 3e0a 37020 . 2  |-  A  e.  ( A  u.  { A } )
5 df-suc 5445 . 2  |-  suc  A  =  ( A  u.  { A } )
64, 5eleqtrri 2509 1  |-  A  e. 
suc  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1868   _Vcvv 3081    u. cun 3434   {csn 3996   suc csuc 5441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-un 3441  df-in 3443  df-ss 3450  df-sn 3997  df-suc 5445
This theorem is referenced by: (None)
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