Theorem sucidVD 42802

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 6377 is sucidVD 42802 without virtual deductions and was automatically derived from sucidVD 42802.

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 4608	. . 3 ⊢ 𝐴 ∈ {𝐴}
3		elun2 4123	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴}))
4	2, 3	e0a 42702	. 2 ⊢ 𝐴 ∈ (𝐴 ∪ {𝐴})
5		df-suc 6302	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	4, 5	eleqtrri 2836	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2105  Vcvv 3441   ∪ cun 3895  {csn 4572  suc csuc 6298

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 3902  df-in 3904  df-ss 3914  df-sn 4573  df-suc 6302

This theorem is referenced by: (None)