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Theorem sucidALTVD 39420
Description: A set belongs to its successor. Alternate proof of sucid 5842. 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 39421 is sucidALTVD 39420 without virtual deductions and was automatically derived from sucidALTVD 39420. 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 ∪ {𝐴}), which unifies with df-suc 5767, 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 𝑥𝐴𝑦𝐴(𝑥𝑦𝑥 = 𝑦𝑦𝑥) is a semantic variation of the theorem (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 𝑦𝐴(𝑥𝑦𝑥 = 𝑦𝑦𝑥))), which unifies with the set.mm reference definition (axiom) dford2 8555.
 h1:: ⊢ 𝐴 ∈ V 2:1: ⊢ 𝐴 ∈ {𝐴} 3:2: ⊢ 𝐴 ∈ ({𝐴} ∪ 𝐴) 4:: ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴) qed:3,4: ⊢ 𝐴 ∈ suc 𝐴
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
sucidALTVD.1 𝐴 ∈ V
Assertion
Ref Expression
sucidALTVD 𝐴 ∈ suc 𝐴

Proof of Theorem sucidALTVD
StepHypRef Expression
1 sucidALTVD.1 . . . 4 𝐴 ∈ V
21snid 4241 . . 3 𝐴 ∈ {𝐴}
3 elun1 3813 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
42, 3e0a 39316 . 2 𝐴 ∈ ({𝐴} ∪ 𝐴)
5 df-suc 5767 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
65equncomi 3792 . 2 suc 𝐴 = ({𝐴} ∪ 𝐴)
74, 6eleqtrri 2729 1 𝐴 ∈ suc 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605  {csn 4210  suc csuc 5763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-sn 4211  df-suc 5767 This theorem is referenced by: (None)
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