Theorem snelpw

Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998)

	Ref	Expression
Hypothesis	snelpw.1	$$⊢A\in \mathrm{V}$$
Assertion	snelpw	$$⊢A\in B\leftrightarrow \left\{A\right\}\in 𝒫B$$

Proof

Step	Hyp	Ref	Expression
1		snelpw.1	$$⊢A\in \mathrm{V}$$
2	1	snss	$$⊢A\in B\leftrightarrow \left\{A\right\}\subseteq B$$
3		snex	$$⊢\left\{A\right\}\in \mathrm{V}$$
4	3	elpw	$$⊢\left\{A\right\}\in 𝒫B\leftrightarrow \left\{A\right\}\subseteq B$$
5	2 4	bitr4i	$$⊢A\in B\leftrightarrow \left\{A\right\}\in 𝒫B$$