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 ( 𝐴𝐵 ↔ { 𝐴 } ∈ 𝒫 𝐵 )

Proof

Step Hyp Ref Expression
1 snelpw.ex 𝐴 ∈ V
2 snelpwg ( 𝐴 ∈ V → ( 𝐴𝐵 ↔ { 𝐴 } ∈ 𝒫 𝐵 ) )
3 1 2 ax-mp ( 𝐴𝐵 ↔ { 𝐴 } ∈ 𝒫 𝐵 )