Metamath Proof Explorer

Theorem sucidVD

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

h1::	\|- A e. _V
2:1:	\|- A e. { A }
3:2:	\|- A e. ( A u. { A } )
4::	\|- suc A = ( A u. { A } )
qed:3,4:	\|- A e. suc A

(Contributed by Alan Sare, 18-Feb-2012) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sucidVD.1	⊢ 𝐴 ∈ V
	Assertion	sucidVD	⊢ 𝐴 ∈ suc 𝐴

Proof

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