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 ) |
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 ) |