Theorem 0uhgrrusgr

Description: The null graph as hypergraph is a k-regular simple graph for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018) (Revised by AV, 26-Dec-2020)

	Ref	Expression
Assertion	0uhgrrusgr	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ∀ 𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘 )

Proof

Step	Hyp	Ref	Expression
1		uhgr0vb	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ UHGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
2	1	biimpd	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ UHGraph → ( iEdg ‘ 𝐺 ) = ∅ ) )
3	2	ex	⊢ ( 𝐺 ∈ UHGraph → ( ( Vtx ‘ 𝐺 ) = ∅ → ( 𝐺 ∈ UHGraph → ( iEdg ‘ 𝐺 ) = ∅ ) ) )
4	3	pm2.43a	⊢ ( 𝐺 ∈ UHGraph → ( ( Vtx ‘ 𝐺 ) = ∅ → ( iEdg ‘ 𝐺 ) = ∅ ) )
5	4	imp	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( iEdg ‘ 𝐺 ) = ∅ )
6		0vtxrusgr	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → ∀ 𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘 )
7	5 6	mpd3an3	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ∀ 𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘 )