Theorem uhgr0edg0rgr

Description: A hypergraph is 0-regular if it has no edges. (Contributed by AV, 19-Dec-2020)

	Ref	Expression
Assertion	uhgr0edg0rgr	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Edg ‘ 𝐺 ) = ∅ ) → 𝐺 RegGraph 0 )

Proof

Step	Hyp	Ref	Expression
1		uhgriedg0edg0	⊢ ( 𝐺 ∈ UHGraph → ( ( Edg ‘ 𝐺 ) = ∅ ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
2	1	biimpa	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Edg ‘ 𝐺 ) = ∅ ) → ( iEdg ‘ 𝐺 ) = ∅ )
3		0edg0rgr	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 RegGraph 0 )
4	2 3	syldan	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Edg ‘ 𝐺 ) = ∅ ) → 𝐺 RegGraph 0 )