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	$$⊢E=\mathrm{Edg}\left(G\right)$$
Assertion	usgredgne	$$⊢\left(G\in \mathrm{USGraph}\wedge \left\{M,N\right\}\in E\right)\to M\ne N$$

Proof

Step	Hyp	Ref	Expression
1		usgredgne.v	$$⊢E=\mathrm{Edg}\left(G\right)$$
2		usgrumgr	$$⊢G\in \mathrm{USGraph}\to G\in \mathrm{UMGraph}$$
3	1	umgredgne	$$⊢\left(G\in \mathrm{UMGraph}\wedge \left\{M,N\right\}\in E\right)\to M\ne N$$
4	2 3	sylan	$$⊢\left(G\in \mathrm{USGraph}\wedge \left\{M,N\right\}\in E\right)\to M\ne N$$