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Theorem sucidVD 39524
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 5917 is sucidVD 39524 without virtual deductions and was automatically derived from sucidVD 39524.
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 4316 . . 3 𝐴 ∈ {𝐴}
3 elun2 3889 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴}))
42, 3e0a 39418 . 2 𝐴 ∈ (𝐴 ∪ {𝐴})
5 df-suc 5842 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
64, 5eleqtrri 2802 1 𝐴 ∈ suc 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2103  Vcvv 3304  cun 3678  {csn 4285  suc csuc 5838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-v 3306  df-un 3685  df-in 3687  df-ss 3694  df-sn 4286  df-suc 5842
This theorem is referenced by: (None)
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