Theorem frgr0vb

Description: Any null graph (without vertices and edges) is a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017) (Revised by AV, 29-Mar-2021)

	Ref	Expression
Assertion	frgr0vb	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ FriendGraph )

Proof

Step	Hyp	Ref	Expression
1		frgr0v	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ FriendGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
2	1	biimp3ar	⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ FriendGraph )