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Theorem sucidVD 44892
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 6466 is sucidVD 44892 without virtual deductions and was automatically derived from sucidVD 44892.
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 4662 . . 3 𝐴 ∈ {𝐴}
3 elun2 4183 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴}))
42, 3e0a 44792 . 2 𝐴 ∈ (𝐴 ∪ {𝐴})
5 df-suc 6390 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
64, 5eleqtrri 2840 1 𝐴 ∈ suc 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  Vcvv 3480  cun 3949  {csn 4626  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-suc 6390
This theorem is referenced by: (None)
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