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Theorem sucidVD 42821
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 6383 is sucidVD 42821 without virtual deductions and was automatically derived from sucidVD 42821.
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 4609 . . 3 𝐴 ∈ {𝐴}
3 elun2 4124 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴}))
42, 3e0a 42721 . 2 𝐴 ∈ (𝐴 ∪ {𝐴})
5 df-suc 6308 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
64, 5eleqtrri 2836 1 𝐴 ∈ suc 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  Vcvv 3441  cun 3896  {csn 4573  suc csuc 6304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-un 3903  df-in 3905  df-ss 3915  df-sn 4574  df-suc 6308
This theorem is referenced by: (None)
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