Theorem usgrnloop0

Description: A simple graph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017) (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	usgrnloop0	$$⊢G\in \mathrm{USGraph}\to \left\{x\in \mathrm{dom}E|E\left(x\right)=\left\{U\right\}\right\}=\varnothing$$

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	umgrnloop0	$$⊢G\in \mathrm{UMGraph}\to \left\{x\in \mathrm{dom}E|E\left(x\right)=\left\{U\right\}\right\}=\varnothing$$
4	2 3	syl	$$⊢G\in \mathrm{USGraph}\to \left\{x\in \mathrm{dom}E|E\left(x\right)=\left\{U\right\}\right\}=\varnothing$$