Metamath Proof Explorer


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 𝐴𝐴 ∈ V )
4 1 3 impbii ( 𝐴 ∈ V ↔ 𝐴 I 𝐴 )