Theorem dm0

Description: The domain of the empty set is empty. Part of Theorem 3.8(v) of Monk1 p. 36. (Contributed by NM, 4-Jul-1994) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	dm0	$$⊢\mathrm{dom}\varnothing =\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		noel	$$⊢\neg ⟨x,y⟩\in \varnothing$$
2	1	nex	$$⊢\neg \exists y⟨x,y⟩\in \varnothing$$
3		vex	$$⊢x\in \mathrm{V}$$
4	3	eldm2	$$⊢x\in \mathrm{dom}\varnothing \leftrightarrow \exists y⟨x,y⟩\in \varnothing$$
5	2 4	mtbir	$$⊢\neg x\in \mathrm{dom}\varnothing$$
6	5	nel0	$$⊢\mathrm{dom}\varnothing =\varnothing$$