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Theorem sucidALTVD 28691
Description: A set belongs to its successor. Alternate proof of sucid 4620. 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. sucidALT 28692 is sucidALTVD 28691 without virtual deductions and was automatically derived from sucidALTVD 28691. This proof illustrates that completeusersproof.cmd will generate a Metamath proof from any User's Proof which is "conventional" in the sense that no step is a virtual deduction, provided that all necessary unification theorems and transformation deductions are in set.mm. completeusersproof.cmd automatically converts such a conventional proof into a Virtual Deduction proof for which each step happens to be a 0-virtual hypothesis virtual deduction. The user does not need to search for reference theorem labels or deduction labels nor does he(she) need to use theorems and deductions which unify with reference theorems and deductions in set.mm. All that is necessary is that each theorem or deduction of the User's Proof unifies with some reference theorem or deduction in set.mm or is a semantic variation of some theorem or deduction which unifies with some reference theorem or deduction in set.mm. The definition of "semantic variation" has not been precisely defined. If it is obvious that a theorem or deduction has the same meaning as another theorem or deduction, then it is a semantic variation of the latter theorem or deduction. For example, step 4 of the User's Proof is a semantic variation of the definition (axiom)  suc  A  =  ( A  u.  { A } ), which unifies with df-suc 4547, a reference definition (axiom) in set.mm. Also, a theorem or deduction is said to be a semantic variation of another theorem or deduction if it is obvious upon cursory inspection that it has the same meaning as a weaker form of the latter theorem or deduction. For example, the deduction  Ord  A infers  A. x  e.  A A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x ) is a semantic variation of the theorem  ( Ord  A  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ) ), which unifies with the set.mm reference definition (axiom) dford2 7531.
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
sucidALTVD.1  |-  A  e. 
_V
Assertion
Ref Expression
sucidALTVD  |-  A  e. 
suc  A

Proof of Theorem sucidALTVD
StepHypRef Expression
1 sucidALTVD.1 . . . 4  |-  A  e. 
_V
21snid 3801 . . 3  |-  A  e. 
{ A }
3 elun1 3474 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( { A }  u.  A
) )
42, 3e0_ 28593 . 2  |-  A  e.  ( { A }  u.  A )
5 df-suc 4547 . . 3  |-  suc  A  =  ( A  u.  { A } )
65equncomi 3453 . 2  |-  suc  A  =  ( { A }  u.  A )
74, 6eleqtrri 2477 1  |-  A  e. 
suc  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   _Vcvv 2916    u. cun 3278   {csn 3774   suc csuc 4543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-in 3287  df-ss 3294  df-sn 3780  df-suc 4547
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