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	$$⊢A\in \mathrm{V}$$
Assertion	snelpw	$$⊢A\in B\leftrightarrow \left\{A\right\}\in 𝒫B$$

Proof

Step	Hyp	Ref	Expression
1		snelpw.ex	$$⊢A\in \mathrm{V}$$
2		snelpwg	$$⊢A\in \mathrm{V}\to \left(A\in B\leftrightarrow \left\{A\right\}\in 𝒫B\right)$$
3	1 2	ax-mp	$$⊢A\in B\leftrightarrow \left\{A\right\}\in 𝒫B$$