Theorem usgrpredgv

Description: An edge of a simple graph always connects two vertices. Analogue of usgredgprv . (Contributed by Alexander van der Vekens, 7-Oct-2017) (Revised by AV, 9-Jan-2020) (Revised by AV, 23-Oct-2020) (Proof shortened by AV, 27-Nov-2020)

	Ref	Expression
Hypotheses	usgredgppr.e	|- E = ( Edg ` G )
	usgrpredgv.v	|- V = ( Vtx ` G )
Assertion	usgrpredgv	|- ( ( G e. USGraph /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )

Proof

Step	Hyp	Ref	Expression
1		usgredgppr.e	|- E = ( Edg ` G )
2		usgrpredgv.v	|- V = ( Vtx ` G )
3		usgrumgr	|- ( G e. USGraph -> G e. UMGraph )
4	2 1	umgrpredgv	|- ( ( G e. UMGraph /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )
5	3 4	sylan	|- ( ( G e. USGraph /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )