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Theorem sucidALTVD 44212
Description: A set belongs to its successor. Alternate proof of sucid 6440. 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 44213 is sucidALTVD 44212 without virtual deductions and was automatically derived from sucidALTVD 44212. 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 6364, 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 9617.
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 4659 . . 3 𝐴 ∈ {𝐴}
3 elun1 4171 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ ({𝐴} ∪ 𝐴))
42, 3e0a 44114 . 2 𝐴 ∈ ({𝐴} ∪ 𝐴)
5 df-suc 6364 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
65equncomi 4150 . 2 suc 𝐴 = ({𝐴} ∪ 𝐴)
74, 6eleqtrri 2826 1 𝐴 ∈ suc 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  Vcvv 3468  cun 3941  {csn 4623  suc csuc 6360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-un 3948  df-in 3950  df-ss 3960  df-sn 4624  df-suc 6364
This theorem is referenced by: (None)
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