Theorem dfnbgr5

Description: Alternate definition of the (open) neighborhood of a vertex as a semiclosed neighborhood without itself. (Contributed by AV, 16-May-2025)

	Ref	Expression
Hypotheses	dfsclnbgr2.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	dfsclnbgr2.s	⊢ 𝑆 = { 𝑛 ∈ 𝑉 ∣ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 }
	dfsclnbgr2.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	dfnbgr5	⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑆 ∖ { 𝑁 } ) )

Proof

Step	Hyp	Ref	Expression
1		dfsclnbgr2.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		dfsclnbgr2.s	⊢ 𝑆 = { 𝑛 ∈ 𝑉 ∣ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 }
3		dfsclnbgr2.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
4		rabdif	⊢ ( { 𝑛 ∈ 𝑉 ∣ ∃ 𝑒 ∈ 𝐸 ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) } ∖ { 𝑁 } ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) }
5	4	eqcomi	⊢ { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) } = ( { 𝑛 ∈ 𝑉 ∣ ∃ 𝑒 ∈ 𝐸 ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) } ∖ { 𝑁 } )
6	1 3	dfnbgr2	⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) } )
7	1 2 3	dfsclnbgr2	⊢ ( 𝑁 ∈ 𝑉 → 𝑆 = { 𝑛 ∈ 𝑉 ∣ ∃ 𝑒 ∈ 𝐸 ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) } )
8	7	difeq1d	⊢ ( 𝑁 ∈ 𝑉 → ( 𝑆 ∖ { 𝑁 } ) = ( { 𝑛 ∈ 𝑉 ∣ ∃ 𝑒 ∈ 𝐸 ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) } ∖ { 𝑁 } ) )
9	5 6 8	3eqtr4a	⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑆 ∖ { 𝑁 } ) )