Theorem 0thinc

Description: The empty category (see 0cat ) is thin. (Contributed by Zhi Wang, 17-Sep-2024)

	Ref	Expression
Assertion	0thinc	$$⊢\varnothing \in \mathrm{ThinCat}$$

Proof

Step	Hyp	Ref	Expression
1		0ex	$$⊢\varnothing \in \mathrm{V}$$
2		base0	$$⊢\varnothing ={\mathrm{Base}}_{\varnothing }$$
3		0thincg	$$⊢\left(\varnothing \in \mathrm{V}\wedge \varnothing ={\mathrm{Base}}_{\varnothing }\right)\to \varnothing \in \mathrm{ThinCat}$$
4	1 2 3	mp2an	$$⊢\varnothing \in \mathrm{ThinCat}$$