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Metamath Proof Explorer

Theorem nbusgrvtx

Description: The set of neighbors of a vertex in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017) (Revised by AV, 26-Oct-2020) (Proof shortened by AV, 27-Nov-2020)

	Ref	Expression
Hypotheses	nbuhgr.v	$⊢ V = Vtx (G)$
	nbuhgr.e	$⊢ E = Edg (G)$
Assertion	nbusgrvtx	$⊢ (G \in USGraph \land N \in V) \to G NeighbVtx N = \{n \in V \| \{N, n\} \in E\}$

Proof

Step	Hyp	Ref	Expression
1		nbuhgr.v	$⊢ V = Vtx (G)$
2		nbuhgr.e	$⊢ E = Edg (G)$
3		usgrumgr	$⊢ G \in USGraph \to G \in UMGraph$
4	1 2	nbumgrvtx	$⊢ (G \in UMGraph \land N \in V) \to G NeighbVtx N = \{n \in V \| \{N, n\} \in E\}$
5	3 4	sylan	$⊢ (G \in USGraph \land N \in V) \to G NeighbVtx N = \{n \in V \| \{N, n\} \in E\}$