Theorem sucidVD 44912

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 6390 is sucidVD 44912 without virtual deductions and was automatically derived from sucidVD 44912.

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 4612	. . 3 ⊢ 𝐴 ∈ {𝐴}
3		elun2 4130	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴}))
4	2, 3	e0a 44812	. 2 ⊢ 𝐴 ∈ (𝐴 ∪ {𝐴})
5		df-suc 6312	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	4, 5	eleqtrri 2830	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2111  Vcvv 3436   ∪ cun 3895  {csn 4573  suc csuc 6308

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-ss 3914  df-sn 4574  df-suc 6312

This theorem is referenced by: (None)