Theorem relsng

Description: A singleton is a relation iff it is a singleton on an ordered pair. (Contributed by NM, 24-Sep-2013) (Revised by BJ, 12-Feb-2022)

	Ref	Expression
Assertion	relsng	$$⊢A\in V\to \left(\mathrm{Rel}\left\{A\right\}\leftrightarrow A\in \left(\mathrm{V}\times \mathrm{V}\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		snssg	$$⊢A\in V\to \left(A\in \left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \left\{A\right\}\subseteq \mathrm{V}\times \mathrm{V}\right)$$
2		df-rel	$$⊢\mathrm{Rel}\left\{A\right\}\leftrightarrow \left\{A\right\}\subseteq \mathrm{V}\times \mathrm{V}$$
3	1 2	syl6rbbr	$$⊢A\in V\to \left(\mathrm{Rel}\left\{A\right\}\leftrightarrow A\in \left(\mathrm{V}\times \mathrm{V}\right)\right)$$