Theorem umgredgne

Description: An edge of a multigraph always connects two different vertices. Analogue of umgrnloopv resp. umgrnloop . (Contributed by AV, 27-Nov-2020)

	Ref	Expression
Hypothesis	umgredgne.v	|- E = ( Edg ` G )
Assertion	umgredgne	|- ( ( G e. UMGraph /\ { M , N } e. E ) -> M =/= N )

Proof

Step	Hyp	Ref	Expression
1		umgredgne.v	|- E = ( Edg ` G )
2	1	eleq2i	|- ( { M , N } e. E <-> { M , N } e. ( Edg ` G ) )
3		edgumgr	|- ( ( G e. UMGraph /\ { M , N } e. ( Edg ` G ) ) -> ( { M , N } e. ~P ( Vtx ` G ) /\ ( # ` { M , N } ) = 2 ) )
4	2 3	sylan2b	|- ( ( G e. UMGraph /\ { M , N } e. E ) -> ( { M , N } e. ~P ( Vtx ` G ) /\ ( # ` { M , N } ) = 2 ) )
5		eqid	|- { M , N } = { M , N }
6	5	hashprdifel	|- ( ( # ` { M , N } ) = 2 -> ( M e. { M , N } /\ N e. { M , N } /\ M =/= N ) )
7	6	simp3d	|- ( ( # ` { M , N } ) = 2 -> M =/= N )
8	4 7	simpl2im	|- ( ( G e. UMGraph /\ { M , N } e. E ) -> M =/= N )