Theorem clnbgr0edg

Description: In an empty graph (with no edges), all closed neighborhoods consists of a single vertex. (Contributed by AV, 10-May-2025)

	Ref	Expression
Assertion	clnbgr0edg	⊢ ( ( ( Edg ‘ 𝐺 ) = ∅ ∧ 𝐾 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ClNeighbVtx 𝐾 ) = { 𝐾 } )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2	1	dfclnbgr4	⊢ ( 𝐾 ∈ ( Vtx ‘ 𝐺 ) → ( 𝐺 ClNeighbVtx 𝐾 ) = ( { 𝐾 } ∪ ( 𝐺 NeighbVtx 𝐾 ) ) )
3	2	adantl	⊢ ( ( ( Edg ‘ 𝐺 ) = ∅ ∧ 𝐾 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ClNeighbVtx 𝐾 ) = ( { 𝐾 } ∪ ( 𝐺 NeighbVtx 𝐾 ) ) )
4		nbgr0edg	⊢ ( ( Edg ‘ 𝐺 ) = ∅ → ( 𝐺 NeighbVtx 𝐾 ) = ∅ )
5	4	adantr	⊢ ( ( ( Edg ‘ 𝐺 ) = ∅ ∧ 𝐾 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 NeighbVtx 𝐾 ) = ∅ )
6	5	uneq2d	⊢ ( ( ( Edg ‘ 𝐺 ) = ∅ ∧ 𝐾 ∈ ( Vtx ‘ 𝐺 ) ) → ( { 𝐾 } ∪ ( 𝐺 NeighbVtx 𝐾 ) ) = ( { 𝐾 } ∪ ∅ ) )
7		un0	⊢ ( { 𝐾 } ∪ ∅ ) = { 𝐾 }
8	7	a1i	⊢ ( ( ( Edg ‘ 𝐺 ) = ∅ ∧ 𝐾 ∈ ( Vtx ‘ 𝐺 ) ) → ( { 𝐾 } ∪ ∅ ) = { 𝐾 } )
9	3 6 8	3eqtrd	⊢ ( ( ( Edg ‘ 𝐺 ) = ∅ ∧ 𝐾 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ClNeighbVtx 𝐾 ) = { 𝐾 } )