Theorem sucidVD 39108

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 5804 is sucidVD 39108 without virtual deductions and was automatically derived from sucidVD 39108.

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 4208	. . 3 ⊢ 𝐴 ∈ {𝐴}
3		elun2 3781	. . 3 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ (𝐴 ∪ {𝐴}))
4	2, 3	e0a 38999	. 2 ⊢ 𝐴 ∈ (𝐴 ∪ {𝐴})
5		df-suc 5729	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
6	4, 5	eleqtrri 2700	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572  {csn 4177  suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-suc 5729

This theorem is referenced by: (None)