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Theorem sucidVD 38130
Description: A set belongs to its successor. 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. sucid 5721 is sucidVD 38130 without virtual deductions and was automatically derived from sucidVD 38130.
 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
sucidVD.1 𝐴 ∈ V
Assertion
Ref Expression
sucidVD 𝐴 ∈ suc 𝐴

Proof of Theorem sucidVD
StepHypRef Expression
1 sucidVD.1 . . . 4 𝐴 ∈ V
21snid 4155 . . 3 𝐴 ∈ {𝐴}
3 elun2 3743 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴}))
42, 3e0a 38020 . 2 𝐴 ∈ (𝐴 ∪ {𝐴})
5 df-suc 5646 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
64, 5eleqtrri 2687 1 𝐴 ∈ suc 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538  {csn 4125  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-suc 5646 This theorem is referenced by: (None)
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