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Theorem sucidALTVD 31908
Description: A set belongs to its successor. Alternate proof of sucid 4898. 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. sucidALT 31909 is sucidALTVD 31908 without virtual deductions and was automatically derived from sucidALTVD 31908. 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 4825, 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 7929.
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 4005 . . 3  |-  A  e. 
{ A }
3 elun1 3623 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( { A }  u.  A
) )
42, 3e0a 31807 . 2  |-  A  e.  ( { A }  u.  A )
5 df-suc 4825 . . 3  |-  suc  A  =  ( A  u.  { A } )
65equncomi 3602 . 2  |-  suc  A  =  ( { A }  u.  A )
74, 6eleqtrri 2538 1  |-  A  e. 
suc  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758   _Vcvv 3070    u. cun 3426   {csn 3977   suc csuc 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3072  df-un 3433  df-in 3435  df-ss 3442  df-sn 3978  df-suc 4825
This theorem is referenced by: (None)
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