Theorem relsn

Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013)

	Ref	Expression
Hypothesis	relsn.1	$$⊢A\in \mathrm{V}$$
Assertion	relsn	$$⊢\mathrm{Rel}\left\{A\right\}\leftrightarrow A\in \left(\mathrm{V}\times \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		relsn.1	$$⊢A\in \mathrm{V}$$
2		relsng	$$⊢A\in \mathrm{V}\to \left(\mathrm{Rel}\left\{A\right\}\leftrightarrow A\in \left(\mathrm{V}\times \mathrm{V}\right)\right)$$
3	1 2	ax-mp	$$⊢\mathrm{Rel}\left\{A\right\}\leftrightarrow A\in \left(\mathrm{V}\times \mathrm{V}\right)$$