Theorem clnbgr0vtx

Description: In a null graph (with no vertices), all closed neighborhoods are empty. (Contributed by AV, 15-Nov-2020)

	Ref	Expression
Assertion	clnbgr0vtx	⊢ ( ( Vtx ‘ 𝐺 ) = ∅ → ( 𝐺 ClNeighbVtx 𝐾 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		nel02	⊢ ( ( Vtx ‘ 𝐺 ) = ∅ → ¬ 𝐾 ∈ ( Vtx ‘ 𝐺 ) )
2		df-nel	⊢ ( 𝐾 ∉ ( Vtx ‘ 𝐺 ) ↔ ¬ 𝐾 ∈ ( Vtx ‘ 𝐺 ) )
3	1 2	sylibr	⊢ ( ( Vtx ‘ 𝐺 ) = ∅ → 𝐾 ∉ ( Vtx ‘ 𝐺 ) )
4		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
5	4	clnbgrnvtx0	⊢ ( 𝐾 ∉ ( Vtx ‘ 𝐺 ) → ( 𝐺 ClNeighbVtx 𝐾 ) = ∅ )
6	3 5	syl	⊢ ( ( Vtx ‘ 𝐺 ) = ∅ → ( 𝐺 ClNeighbVtx 𝐾 ) = ∅ )