Theorem umgredgne

Description: An edge of a multigraph always connects two different vertices. Analogue of umgrnloopv resp. umgrnloop . (Contributed by AV, 27-Nov-2020)

	Ref	Expression
Hypothesis	umgredgne.v	$$⊢E=\mathrm{Edg}\left(G\right)$$
Assertion	umgredgne	$$⊢\left(G\in \mathrm{UMGraph}\wedge \left\{M,N\right\}\in E\right)\to M\ne N$$

Proof

Step	Hyp	Ref	Expression
1		umgredgne.v	$$⊢E=\mathrm{Edg}\left(G\right)$$
2	1	eleq2i	$$⊢\left\{M,N\right\}\in E\leftrightarrow \left\{M,N\right\}\in \mathrm{Edg}\left(G\right)$$
3		edgumgr	$$⊢\left(G\in \mathrm{UMGraph}\wedge \left\{M,N\right\}\in \mathrm{Edg}\left(G\right)\right)\to \left(\left\{M,N\right\}\in 𝒫\mathrm{Vtx}\left(G\right)\wedge \left|\left\{M,N\right\}\right|=2\right)$$
4	2 3	sylan2b	$$⊢\left(G\in \mathrm{UMGraph}\wedge \left\{M,N\right\}\in E\right)\to \left(\left\{M,N\right\}\in 𝒫\mathrm{Vtx}\left(G\right)\wedge \left|\left\{M,N\right\}\right|=2\right)$$
5		eqid	$$⊢\left\{M,N\right\}=\left\{M,N\right\}$$
6	5	hashprdifel	$$⊢\left|\left\{M,N\right\}\right|=2\to \left(M\in \left\{M,N\right\}\wedge N\in \left\{M,N\right\}\wedge M\ne N\right)$$
7	6	simp3d	$$⊢\left|\left\{M,N\right\}\right|=2\to M\ne N$$
8	4 7	simpl2im	$$⊢\left(G\in \mathrm{UMGraph}\wedge \left\{M,N\right\}\in E\right)\to M\ne N$$