Theorem uhgredgn0

Description: An edge of a hypergraph is a nonempty subset of vertices. (Contributed by AV, 28-Nov-2020)

	Ref	Expression
Assertion	uhgredgn0	|- ( ( G e. UHGraph /\ E e. ( Edg ` G ) ) -> E e. ( ~P ( Vtx ` G ) \ { (/) } ) )

Proof

Step	Hyp	Ref	Expression
1		edgval	|- ( Edg ` G ) = ran ( iEdg ` G )
2		eqid	|- ( Vtx ` G ) = ( Vtx ` G )
3		eqid	|- ( iEdg ` G ) = ( iEdg ` G )
4	2 3	uhgrf	|- ( G e. UHGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) --> ( ~P ( Vtx ` G ) \ { (/) } ) )
5	4	frnd	|- ( G e. UHGraph -> ran ( iEdg ` G ) C_ ( ~P ( Vtx ` G ) \ { (/) } ) )
6	1 5	eqsstrid	|- ( G e. UHGraph -> ( Edg ` G ) C_ ( ~P ( Vtx ` G ) \ { (/) } ) )
7	6	sselda	|- ( ( G e. UHGraph /\ E e. ( Edg ` G ) ) -> E e. ( ~P ( Vtx ` G ) \ { (/) } ) )