Theorem wlk0prc

Description: There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018) (Revised by AV, 5-Mar-2021)

	Ref	Expression
Assertion	wlk0prc	$$⊢\left(S\notin \mathrm{V}\wedge \mathrm{Vtx}\left(S\right)=\mathrm{Vtx}\left(G\right)\right)\to \mathrm{Walks}\left(G\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		eqcom	$$⊢\mathrm{Vtx}\left(S\right)=\mathrm{Vtx}\left(G\right)\leftrightarrow \mathrm{Vtx}\left(G\right)=\mathrm{Vtx}\left(S\right)$$
2	1	biimpi	$$⊢\mathrm{Vtx}\left(S\right)=\mathrm{Vtx}\left(G\right)\to \mathrm{Vtx}\left(G\right)=\mathrm{Vtx}\left(S\right)$$
3		vtxvalprc	$$⊢S\notin \mathrm{V}\to \mathrm{Vtx}\left(S\right)=\varnothing$$
4	2 3	sylan9eqr	$$⊢\left(S\notin \mathrm{V}\wedge \mathrm{Vtx}\left(S\right)=\mathrm{Vtx}\left(G\right)\right)\to \mathrm{Vtx}\left(G\right)=\varnothing$$
5		g0wlk0	$$⊢\mathrm{Vtx}\left(G\right)=\varnothing \to \mathrm{Walks}\left(G\right)=\varnothing$$
6	4 5	syl	$$⊢\left(S\notin \mathrm{V}\wedge \mathrm{Vtx}\left(S\right)=\mathrm{Vtx}\left(G\right)\right)\to \mathrm{Walks}\left(G\right)=\varnothing$$