Theorem usgredgne

Description: An edge of a simple graph always connects two different vertices. Analogue of usgrnloopv resp. usgrnloop . (Contributed by Alexander van der Vekens, 2-Sep-2017) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 27-Nov-2020)

	Ref	Expression
Hypothesis	usgredgne.v	|- E = ( Edg ` G )
Assertion	usgredgne	|- ( ( G e. USGraph /\ { M , N } e. E ) -> M =/= N )

Proof

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