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Theorem sucidVD 39422
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 5842 is sucidVD 39422 without virtual deductions and was automatically derived from sucidVD 39422.
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 4241 . . 3 𝐴 ∈ {𝐴}
3 elun2 3814 . . 3 (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴}))
42, 3e0a 39316 . 2 𝐴 ∈ (𝐴 ∪ {𝐴})
5 df-suc 5767 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
64, 5eleqtrri 2729 1 𝐴 ∈ suc 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  Vcvv 3231  cun 3605  {csn 4210  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-sn 4211  df-suc 5767
This theorem is referenced by: (None)
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