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Metamath Proof Explorer

Theorem snelpw

Description: A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998)

Ref Expression
Hypothesis snelpw.ex ⊢ A ∈ V
Assertion snelpw ⊢ A ∈ B ↔ A ∈ 𝒫 B

Proof

Step Hyp Ref Expression
1 snelpw.ex ⊢ A ∈ V
2 snelpwg ⊢ A ∈ V → A ∈ B ↔ A ∈ 𝒫 B
3 1 2 ax-mp ⊢ A ∈ B ↔ A ∈ 𝒫 B