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Metamath Proof Explorer

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	$⊢ (S \notin V \land Vtx (S) = Vtx (G)) \to Walks (G) = \emptyset$

Proof

Step	Hyp	Ref	Expression
1		eqcom	$⊢ Vtx (S) = Vtx (G) \leftrightarrow Vtx (G) = Vtx (S)$
2	1	biimpi	$⊢ Vtx (S) = Vtx (G) \to Vtx (G) = Vtx (S)$
3		vtxvalprc	$⊢ S \notin V \to Vtx (S) = \emptyset$
4	2 3	sylan9eqr	$⊢ (S \notin V \land Vtx (S) = Vtx (G)) \to Vtx (G) = \emptyset$
5		g0wlk0	$⊢ Vtx (G) = \emptyset \to Walks (G) = \emptyset$
6	4 5	syl	$⊢ (S \notin V \land Vtx (S) = Vtx (G)) \to Walks (G) = \emptyset$