Theorem sucidVD 42445

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 6342 is sucidVD 42445 without virtual deductions and was automatically derived from sucidVD 42445.

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

Step	Hyp	Ref	Expression
1		sucidVD.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	snid 4602	. . 3 ⊢ 𝐴 ∈ {𝐴}
3		elun2 4115	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴}))
4	2, 3	e0a 42345	. 2 ⊢ 𝐴 ∈ (𝐴 ∪ {𝐴})
5		df-suc 6269	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	4, 5	eleqtrri 2839	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2109  Vcvv 3430   ∪ cun 3889  {csn 4566  suc csuc 6265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-v 3432  df-un 3896  df-in 3898  df-ss 3908  df-sn 4567  df-suc 6269

This theorem is referenced by: (None)