Theorem relsn2

Description: A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013) Make hypothesis an antecedent. (Revised by BJ, 12-Feb-2022)

	Ref	Expression
Assertion	relsn2	$$⊢A\in V\to \left(\mathrm{Rel}\left\{A\right\}\leftrightarrow \mathrm{dom}\left\{A\right\}\ne \varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		relsng	$$⊢A\in V\to \left(\mathrm{Rel}\left\{A\right\}\leftrightarrow A\in \left(\mathrm{V}\times \mathrm{V}\right)\right)$$
2		dmsnn0	$$⊢A\in \left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \mathrm{dom}\left\{A\right\}\ne \varnothing$$
3	1 2	bitrdi	$$⊢A\in V\to \left(\mathrm{Rel}\left\{A\right\}\leftrightarrow \mathrm{dom}\left\{A\right\}\ne \varnothing \right)$$