Theorem snidb

Description: A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004)

	Ref	Expression
Assertion	snidb	⊢ ( 𝐴 ∈ V ↔ 𝐴 ∈ { 𝐴 } )

Proof

Step	Hyp	Ref	Expression
1		snidg	⊢ ( 𝐴 ∈ V → 𝐴 ∈ { 𝐴 } )
2		elex	⊢ ( 𝐴 ∈ { 𝐴 } → 𝐴 ∈ V )
3	1 2	impbii	⊢ ( 𝐴 ∈ V ↔ 𝐴 ∈ { 𝐴 } )