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Metamath Proof Explorer

Theorem clnbgr0vtx

Description: In a null graph (with no vertices), all closed neighborhoods are empty. (Contributed by AV, 15-Nov-2020)

		Ref	Expression
	Assertion	clnbgr0vtx	$⊢ Vtx (G) = \emptyset \to G ClNeighbVtx K = \emptyset$

Step	Hyp	Ref	Expression
1		nel02	$⊢ Vtx (G) = \emptyset \to \neg K \in Vtx (G)$
2		df-nel	$⊢ K \notin Vtx (G) \leftrightarrow \neg K \in Vtx (G)$
3	1 2	sylibr	$⊢ Vtx (G) = \emptyset \to K \notin Vtx (G)$
4		eqid	$⊢ Vtx (G) = Vtx (G)$
5	4	clnbgrnvtx0	$⊢ K \notin Vtx (G) \to G ClNeighbVtx K = \emptyset$
6	3 5	syl	$⊢ Vtx (G) = \emptyset \to G ClNeighbVtx K = \emptyset$