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Theorem sucidALTVD 41433
Description: A set belongs to its successor. Alternate proof of sucid 6257. 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 41434 is sucidALTVD 41433 without virtual deductions and was automatically derived from sucidALTVD 41433. 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 6184, 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 9074.
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 4585 . . 3 𝐴 ∈ {𝐴}
3 elun1 4137 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
42, 3e0a 41335 . 2 𝐴 ∈ ({𝐴} ∪ 𝐴)
5 df-suc 6184 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
65equncomi 4116 . 2 suc 𝐴 = ({𝐴} ∪ 𝐴)
74, 6eleqtrri 2915 1 𝐴 ∈ suc 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2115  Vcvv 3480  cun 3917  {csn 4549  suc csuc 6180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4550  df-suc 6184
This theorem is referenced by: (None)
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