Theorem usgrnloopv

Description: In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

	Ref	Expression
Hypothesis	usgrnloopv.e	|- E = ( iEdg ` G )
Assertion	usgrnloopv	|- ( ( G e. USGraph /\ M e. W ) -> ( ( 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	umgrnloopv	|- ( ( G e. UMGraph /\ M e. W ) -> ( ( E ` X ) = { M , N } -> M =/= N ) )
4	2 3	sylan	|- ( ( G e. USGraph /\ M e. W ) -> ( ( E ` X ) = { M , N } -> M =/= N ) )