Metamath Proof Explorer

Theorem nbusgredgeu

Description: For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017) (Revised by AV, 27-Oct-2020)

		Ref	Expression
	Hypothesis	nbusgredgeu.e	$⊢ E = Edg (G)$
	Assertion	nbusgredgeu	$⊢ (G \in USGraph \land M \in (G NeighbVtx N)) \to \exists! e \in E e = \{M, N\}$

Proof

Step	Hyp	Ref	Expression
1		nbusgredgeu.e	$⊢ E = Edg (G)$
2	1	nbusgreledg	$⊢ G \in USGraph \to (M \in (G NeighbVtx N) \leftrightarrow \{M, N\} \in E)$
3	2	biimpa	$⊢ (G \in USGraph \land M \in (G NeighbVtx N)) \to \{M, N\} \in E$
4		eqeq1	$⊢ e = \{M, N\} \to (e = \{M, N\} \leftrightarrow \{M, N\} = \{M, N\})$
5	4	adantl	$⊢ ((G \in USGraph \land M \in (G NeighbVtx N)) \land e = \{M, N\}) \to (e = \{M, N\} \leftrightarrow \{M, N\} = \{M, N\})$
6		eqidd	$⊢ (G \in USGraph \land M \in (G NeighbVtx N)) \to \{M, N\} = \{M, N\}$
7	3 5 6	rspcedvd	$⊢ (G \in USGraph \land M \in (G NeighbVtx N)) \to \exists e \in E e = \{M, N\}$
8		rmoeq	$⊢ \exists^{*} e \in E e = \{M, N\}$
9		reu5	$⊢ \exists! e \in E e = \{M, N\} \leftrightarrow (\exists e \in E e = \{M, N\} \land \exists^{*} e \in E e = \{M, N\})$
10	7 8 9	sylanblrc	$⊢ (G \in USGraph \land M \in (G NeighbVtx N)) \to \exists! e \in E e = \{M, N\}$