Theorem snelpwi

Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011)

	Ref	Expression
Assertion	snelpwi	$$⊢A\in B\to \left\{A\right\}\in 𝒫B$$

Proof

Step	Hyp	Ref	Expression
1		snssi	$$⊢A\in B\to \left\{A\right\}\subseteq B$$
2		snex	$$⊢\left\{A\right\}\in \mathrm{V}$$
3	2	elpw	$$⊢\left\{A\right\}\in 𝒫B\leftrightarrow \left\{A\right\}\subseteq B$$
4	1 3	sylibr	$$⊢A\in B\to \left\{A\right\}\in 𝒫B$$