Theorem issetid

Description: Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008) (Proof shortened by Andrew Salmon, 27-Aug-2011) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Assertion	issetid	⊢ ( 𝐴 ∈ V ↔ 𝐴 I 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ididg	⊢ ( 𝐴 ∈ V → 𝐴 I 𝐴 )
2		reli	⊢ Rel I
3	2	brrelex1i	⊢ ( 𝐴 I 𝐴 → 𝐴 ∈ V )
4	1 3	impbii	⊢ ( 𝐴 ∈ V ↔ 𝐴 I 𝐴 )