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Metamath Proof Explorer

Theorem pwidb

Description: A class is an element of its powerclass if and only if it is a set. (Contributed by BJ, 31-Dec-2023)

Ref Expression
Assertion pwidb ⊢ A ∈ V ↔ A ∈ 𝒫 A

Proof

Step Hyp Ref Expression
1 pwidg ⊢ A ∈ V → A ∈ 𝒫 A
2 elex ⊢ A ∈ 𝒫 A → A ∈ V
3 1 2 impbii ⊢ A ∈ V ↔ A ∈ 𝒫 A