Theorem nrelv

Description: The universal class is not a relation. (Contributed by Thierry Arnoux, 23-Jan-2022)

	Ref	Expression
Assertion	nrelv	$$⊢\neg \mathrm{Rel}\mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		0ex	$$⊢\varnothing \in \mathrm{V}$$
2		0nelxp	$$⊢\neg \varnothing \in \left(\mathrm{V}\times \mathrm{V}\right)$$
3		nelss	$$⊢\left(\varnothing \in \mathrm{V}\wedge \neg \varnothing \in \left(\mathrm{V}\times \mathrm{V}\right)\right)\to \neg \mathrm{V}\subseteq \mathrm{V}\times \mathrm{V}$$
4	1 2 3	mp2an	$$⊢\neg \mathrm{V}\subseteq \mathrm{V}\times \mathrm{V}$$
5		df-rel	$$⊢\mathrm{Rel}\mathrm{V}\leftrightarrow \mathrm{V}\subseteq \mathrm{V}\times \mathrm{V}$$
6	4 5	mtbir	$$⊢\neg \mathrm{Rel}\mathrm{V}$$