Theorem usgrnloop

Description: In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017) (Proof shortened by Alexander van der Vekens, 20-Mar-2018) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

	Ref	Expression
Hypothesis	usgrnloopv.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
Assertion	usgrnloop	$$⊢G\in \mathrm{USGraph}\to \left(\exists x\in \mathrm{dom}EE\left(x\right)=\left\{M,N\right\}\to M\ne N\right)$$

Proof

Step	Hyp	Ref	Expression
1		usgrnloopv.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
2		usgrumgr	$$⊢G\in \mathrm{USGraph}\to G\in \mathrm{UMGraph}$$
3	1	umgrnloop	$$⊢G\in \mathrm{UMGraph}\to \left(\exists x\in \mathrm{dom}EE\left(x\right)=\left\{M,N\right\}\to M\ne N\right)$$
4	2 3	syl	$$⊢G\in \mathrm{USGraph}\to \left(\exists x\in \mathrm{dom}EE\left(x\right)=\left\{M,N\right\}\to M\ne N\right)$$