Theorem rnsnn0

Description: The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008)

	Ref	Expression
Assertion	rnsnn0	$$⊢A\in \left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \mathrm{ran}\left\{A\right\}\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		dmsnn0	$$⊢A\in \left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \mathrm{dom}\left\{A\right\}\ne \varnothing$$
2		dm0rn0	$$⊢\mathrm{dom}\left\{A\right\}=\varnothing \leftrightarrow \mathrm{ran}\left\{A\right\}=\varnothing$$
3	2	necon3bii	$$⊢\mathrm{dom}\left\{A\right\}\ne \varnothing \leftrightarrow \mathrm{ran}\left\{A\right\}\ne \varnothing$$
4	1 3	bitri	$$⊢A\in \left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \mathrm{ran}\left\{A\right\}\ne \varnothing$$