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Metamath Proof Explorer

Theorem usgredgppr

Description: An edge of a simple graph is a proper pair, i.e. a set containing two different elements (the endvertices of the edge). Analogue of usgredg2 . (Contributed by Alexander van der Vekens, 11-Aug-2017) (Revised by AV, 9-Jan-2020) (Revised by AV, 23-Oct-2020)

		Ref	Expression
	Hypothesis	usgredgppr.e	\|- E = ( Edg ` G )
	Assertion	usgredgppr	\|- ( ( G e. USGraph /\ C e. E ) -> ( # ` C ) = 2 )

Proof

Step	Hyp	Ref	Expression
1		usgredgppr.e	\|- E = ( Edg ` G )
2	1	eleq2i	\|- ( C e. E <-> C e. ( Edg ` G ) )
3		edgusgr	\|- ( ( G e. USGraph /\ C e. ( Edg ` G ) ) -> ( C e. ~P ( Vtx ` G ) /\ ( # ` C ) = 2 ) )
4	2 3	sylan2b	\|- ( ( G e. USGraph /\ C e. E ) -> ( C e. ~P ( Vtx ` G ) /\ ( # ` C ) = 2 ) )
5	4	simprd	\|- ( ( G e. USGraph /\ C e. E ) -> ( # ` C ) = 2 )