Theorem edgusgr

Description: An edge of a simple graph is an unordered pair of vertices. (Contributed by AV, 1-Jan-2020) (Revised by AV, 14-Oct-2020)

	Ref	Expression
Assertion	edgusgr	|- ( ( G e. USGraph /\ E e. ( Edg ` G ) ) -> ( E e. ~P ( Vtx ` G ) /\ ( # ` E ) = 2 ) )

Proof

Step	Hyp	Ref	Expression
1		usgredgss	|- ( G e. USGraph -> ( Edg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
2	1	sselda	|- ( ( G e. USGraph /\ E e. ( Edg ` G ) ) -> E e. { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
3		fveqeq2	|- ( x = E -> ( ( # ` x ) = 2 <-> ( # ` E ) = 2 ) )
4	3	elrab	|- ( E e. { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } <-> ( E e. ~P ( Vtx ` G ) /\ ( # ` E ) = 2 ) )
5	2 4	sylib	|- ( ( G e. USGraph /\ E e. ( Edg ` G ) ) -> ( E e. ~P ( Vtx ` G ) /\ ( # ` E ) = 2 ) )