Metamath Proof Explorer

Theorem nbgrssovtx

Description: The neighbors of a vertex X form a subset of all vertices except the vertex X itself. Stronger version of nbgrssvtx . (Contributed by Alexander van der Vekens, 13-Jul-2018) (Revised by AV, 3-Nov-2020) (Revised by AV, 12-Feb-2022)

		Ref	Expression
	Hypothesis	nbgrssovtx.v	$⊢ V = Vtx (G)$
	Assertion	nbgrssovtx	$⊢ G NeighbVtx X \subseteq V ∖ \{X\}$

Proof

Step	Hyp	Ref	Expression
1		nbgrssovtx.v	$⊢ V = Vtx (G)$
2	1	nbgrisvtx	$⊢ v \in (G NeighbVtx X) \to v \in V$
3		nbgrnself2	$⊢ X \notin (G NeighbVtx X)$
4		df-nel	$⊢ v \notin (G NeighbVtx X) \leftrightarrow \neg v \in (G NeighbVtx X)$
5		neleq1	$⊢ v = X \to (v \notin (G NeighbVtx X) \leftrightarrow X \notin (G NeighbVtx X))$
6	4 5	bitr3id	$⊢ v = X \to (\neg v \in (G NeighbVtx X) \leftrightarrow X \notin (G NeighbVtx X))$
7	3 6	mpbiri	$⊢ v = X \to \neg v \in (G NeighbVtx X)$
8	7	necon2ai	$⊢ v \in (G NeighbVtx X) \to v \neq X$
9		eldifsn	$⊢ v \in (V ∖ \{X\}) \leftrightarrow (v \in V \land v \neq X)$
10	2 8 9	sylanbrc	$⊢ v \in (G NeighbVtx X) \to v \in (V ∖ \{X\})$
11	10	ssriv	$⊢ G NeighbVtx X \subseteq V ∖ \{X\}$