Theorem uhgredgrnv

Description: An edge of a hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018) (Revised by AV, 4-Jun-2021)

	Ref	Expression
Assertion	uhgredgrnv	⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ∧ 𝑁 ∈ 𝐸 ) → 𝑁 ∈ ( Vtx ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		edguhgr	⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ) → 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) )
2		elelpwi	⊢ ( ( 𝑁 ∈ 𝐸 ∧ 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ) → 𝑁 ∈ ( Vtx ‘ 𝐺 ) )
3	2	expcom	⊢ ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) → ( 𝑁 ∈ 𝐸 → 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) )
4	1 3	syl	⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ) → ( 𝑁 ∈ 𝐸 → 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) )
5	4	3impia	⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ∧ 𝑁 ∈ 𝐸 ) → 𝑁 ∈ ( Vtx ‘ 𝐺 ) )