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 ∈ UHGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ) → 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 ∈ UHGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) )
5	4	frnd	⊢ ( 𝐺 ∈ UHGraph → ran ( iEdg ‘ 𝐺 ) ⊆ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) )
6	1 5	eqsstrid	⊢ ( 𝐺 ∈ UHGraph → ( Edg ‘ 𝐺 ) ⊆ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) )
7	6	sselda	⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ) → 𝐸 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) )