Theorem snelpwg

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) Put in closed form and avoid ax-nul . (Revised by BJ, 17-Jan-2025)

	Ref	Expression
Assertion	snelpwg	$$⊢A\in V\to \left(A\in B\leftrightarrow \left\{A\right\}\in 𝒫B\right)$$

Proof

Step	Hyp	Ref	Expression
1		snssg	$$⊢A\in V\to \left(A\in B\leftrightarrow \left\{A\right\}\subseteq B\right)$$
2		snexg	$$⊢A\in V\to \left\{A\right\}\in \mathrm{V}$$
3		elpwg	$$⊢\left\{A\right\}\in \mathrm{V}\to \left(\left\{A\right\}\in 𝒫B\leftrightarrow \left\{A\right\}\subseteq B\right)$$
4	2 3	syl	$$⊢A\in V\to \left(\left\{A\right\}\in 𝒫B\leftrightarrow \left\{A\right\}\subseteq B\right)$$
5	1 4	bitr4d	$$⊢A\in V\to \left(A\in B\leftrightarrow \left\{A\right\}\in 𝒫B\right)$$