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Theorem sucidALTVD 32625
Description: A set belongs to its successor. Alternate proof of sucid 4950. 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 32626 is sucidALTVD 32625 without virtual deductions and was automatically derived from sucidALTVD 32625. 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 4877, 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 8026.
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 4048 . . 3  |-  A  e. 
{ A }
3 elun1 3664 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( { A }  u.  A
) )
42, 3e0a 32524 . 2  |-  A  e.  ( { A }  u.  A )
5 df-suc 4877 . . 3  |-  suc  A  =  ( A  u.  { A } )
65equncomi 3643 . 2  |-  suc  A  =  ( { A }  u.  A )
74, 6eleqtrri 2547 1  |-  A  e. 
suc  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1762   _Vcvv 3106    u. cun 3467   {csn 4020   suc csuc 4873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-un 3474  df-in 3476  df-ss 3483  df-sn 4021  df-suc 4877
This theorem is referenced by: (None)
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