Theorem dmsn0

Description: The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004)

	Ref	Expression
Assertion	dmsn0	$$⊢\mathrm{dom}\left\{\varnothing \right\}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		0nelxp	$$⊢\neg \varnothing \in \left(\mathrm{V}\times \mathrm{V}\right)$$
2		dmsnn0	$$⊢\varnothing \in \left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \mathrm{dom}\left\{\varnothing \right\}\ne \varnothing$$
3	2	necon2bbii	$$⊢\mathrm{dom}\left\{\varnothing \right\}=\varnothing \leftrightarrow \neg \varnothing \in \left(\mathrm{V}\times \mathrm{V}\right)$$
4	1 3	mpbir	$$⊢\mathrm{dom}\left\{\varnothing \right\}=\varnothing$$