Theorem sucidVD 44896

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 6436 is sucidVD 44896 without virtual deductions and was automatically derived from sucidVD 44896.

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 4638	. . 3 ⊢ 𝐴 ∈ {𝐴}
3		elun2 4158	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴}))
4	2, 3	e0a 44796	. 2 ⊢ 𝐴 ∈ (𝐴 ∪ {𝐴})
5		df-suc 6358	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	4, 5	eleqtrri 2833	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2108  Vcvv 3459   ∪ cun 3924  {csn 4601  suc csuc 6354

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-ss 3943  df-sn 4602  df-suc 6358

This theorem is referenced by: (None)