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Theorem sucidALTVD 32449
Description: A set belongs to its successor. Alternate proof of sucid 4915. 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 32450 is sucidALTVD 32449 without virtual deductions and was automatically derived from sucidALTVD 32449. 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) , which unifies with df-suc 4842, 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 infers is a semantic variation of the theorem (OrdA<->(TrA/\A.xe.A A.ye.A(xe.y\/x=y\/ye.x))), which unifies with the set.mm reference definition (axiom) dford2 7963.
 h1:: |-Ae. 2:1: |-Ae.{A} 3:2: |-Ae.({A}u.A) 4:: |-sucA=({A}u.A) qed:3,4: |-Ae.sucA
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
sucidALTVD.1
Assertion
Ref Expression
sucidALTVD

Proof of Theorem sucidALTVD
StepHypRef Expression
1 sucidALTVD.1 . . . 4
21snid 4021 . . 3
3 elun1 3637 . . 3
42, 3e0a 32348 . 2
5 df-suc 4842 . . 3
65equncomi 3616 . 2
74, 6eleqtrri 2541 1
 Colors of variables: wff setvar class Syntax hints:  e.wcel 1758   cvv 3081  u.cun 3440  {csn 3993  succsuc 4838 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 1955  ax-ext 2432 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 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3083  df-un 3447  df-in 3449  df-ss 3456  df-sn 3994  df-suc 4842
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