Theorem sdom0

Description: The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003)

	Ref	Expression
Assertion	sdom0	$$⊢\neg A\prec \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		relsdom	$$⊢\mathrm{Rel}\prec$$
2	1	brrelex1i	$$⊢A\prec \varnothing \to A\in \mathrm{V}$$
3		0domg	$$⊢A\in \mathrm{V}\to \varnothing \preccurlyeq A$$
4	2 3	syl	$$⊢A\prec \varnothing \to \varnothing \preccurlyeq A$$
5		domnsym	$$⊢\varnothing \preccurlyeq A\to \neg A\prec \varnothing$$
6	5	con2i	$$⊢A\prec \varnothing \to \neg \varnothing \preccurlyeq A$$
7	4 6	pm2.65i	$$⊢\neg A\prec \varnothing$$