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Metamath Proof Explorer

Theorem uvtxnbgr

Description: A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017) (Revised by AV, 3-Nov-2020) (Revised by AV, 23-Mar-2021)

		Ref	Expression
	Hypothesis	uvtxnbgr.v	$⊢ V = Vtx (G)$
	Assertion	uvtxnbgr	$⊢ N \in UnivVtx (G) \to G NeighbVtx N = V ∖ \{N\}$

Proof

Step	Hyp	Ref	Expression
1		uvtxnbgr.v	$⊢ V = Vtx (G)$
2	1	nbgrssovtx	$⊢ G NeighbVtx N \subseteq V ∖ \{N\}$
3	2	a1i	$⊢ N \in UnivVtx (G) \to G NeighbVtx N \subseteq V ∖ \{N\}$
4	1	uvtxnbgrss	$⊢ N \in UnivVtx (G) \to V ∖ \{N\} \subseteq G NeighbVtx N$
5	3 4	eqssd	$⊢ N \in UnivVtx (G) \to G NeighbVtx N = V ∖ \{N\}$