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 = ( iEdg ` G )
Assertion	usgrnloop	|- ( G e. USGraph -> ( E. x e. dom E ( E ` x ) = { M , N } -> M =/= N ) )

Proof

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