0vconngr

Description: A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017) (Revised by AV, 15-Feb-2021)

		Ref	Expression
	Assertion	0vconngr	$⊢ (G \in W \land Vtx (G) = \emptyset) \to G \in ConnGraph$

Step	Hyp	Ref	Expression
1		rzal	$⊢ Vtx (G) = \emptyset \to \forall k \in Vtx (G) \forall n \in Vtx (G) \exists f \exists p f (k PathsOn (G) n) p$
2	1	adantl	$⊢ (G \in W \land Vtx (G) = \emptyset) \to \forall k \in Vtx (G) \forall n \in Vtx (G) \exists f \exists p f (k PathsOn (G) n) p$
3		eqid	$⊢ Vtx (G) = Vtx (G)$
4	3	isconngr	$⊢ G \in W \to (G \in ConnGraph \leftrightarrow \forall k \in Vtx (G) \forall n \in Vtx (G) \exists f \exists p f (k PathsOn (G) n) p)$
5	4	adantr	$⊢ (G \in W \land Vtx (G) = \emptyset) \to (G \in ConnGraph \leftrightarrow \forall k \in Vtx (G) \forall n \in Vtx (G) \exists f \exists p f (k PathsOn (G) n) p)$
6	2 5	mpbird	$⊢ (G \in W \land Vtx (G) = \emptyset) \to G \in ConnGraph$

Metamath Proof Explorer