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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 )