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Theorem sucidALTVD 33378
Description: A set belongs to its successor. Alternate proof of sucid 4943. 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 33379 is sucidALTVD 33378 without virtual deductions and was automatically derived from sucidALTVD 33378. 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 4870, 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 8035.
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 4038 . . 3  |-  A  e. 
{ A }
3 elun1 3653 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( { A }  u.  A
) )
42, 3e0a 33277 . 2  |-  A  e.  ( { A }  u.  A )
5 df-suc 4870 . . 3  |-  suc  A  =  ( A  u.  { A } )
65equncomi 3632 . 2  |-  suc  A  =  ( { A }  u.  A )
74, 6eleqtrri 2528 1  |-  A  e. 
suc  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1802   _Vcvv 3093    u. cun 3456   {csn 4010   suc csuc 4866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-v 3095  df-un 3463  df-in 3465  df-ss 3472  df-sn 4011  df-suc 4870
This theorem is referenced by: (None)
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