Theorem frgruhgr0v

Description: Any null graph (without vertices) represented as hypergraph is a friendship graph. (Contributed by AV, 29-Mar-2021)

	Ref	Expression
Assertion	frgruhgr0v	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ FriendGraph )

Proof

Step	Hyp	Ref	Expression
1		uhgr0vb	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ UHGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
2	1	biimpcd	⊢ ( 𝐺 ∈ UHGraph → ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( iEdg ‘ 𝐺 ) = ∅ ) )
3	2	anabsi5	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( iEdg ‘ 𝐺 ) = ∅ )
4		frgr0vb	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ FriendGraph )
5	3 4	mpd3an3	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ FriendGraph )