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 = ( iEdg ` G )
Assertion	usgrnloop0	|- ( G e. USGraph -> { x e. dom E | ( E ` x ) = { U } } = (/) )

Proof

Step	Hyp	Ref	Expression
1		usgrnloopv.e	|- E = ( iEdg ` G )
2		usgrumgr	|- ( G e. USGraph -> G e. UMGraph )
3	1	umgrnloop0	|- ( G e. UMGraph -> { x e. dom E | ( E ` x ) = { U } } = (/) )
4	2 3	syl	|- ( G e. USGraph -> { x e. dom E | ( E ` x ) = { U } } = (/) )