Theorem dfnbgr2

Description: Alternate definition of the neighbors of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 15-Nov-2020) (Revised by AV, 21-Mar-2021)

	Ref	Expression
Hypotheses	nbgrval.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	nbgrval.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	dfnbgr2	⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) } )

Proof

Step	Hyp	Ref	Expression
1		nbgrval.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		nbgrval.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3	1 2	nbgrval	⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 } )
4		prssg	⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑛 ∈ V ) → ( ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) ↔ { 𝑁 , 𝑛 } ⊆ 𝑒 ) )
5	4	elvd	⊢ ( 𝑁 ∈ 𝑉 → ( ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) ↔ { 𝑁 , 𝑛 } ⊆ 𝑒 ) )
6	5	bicomd	⊢ ( 𝑁 ∈ 𝑉 → ( { 𝑁 , 𝑛 } ⊆ 𝑒 ↔ ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) ) )
7	6	rexbidv	⊢ ( 𝑁 ∈ 𝑉 → ( ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ 𝐸 ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) ) )
8	7	rabbidv	⊢ ( 𝑁 ∈ 𝑉 → { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 } = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) } )
9	3 8	eqtrd	⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 ( 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒 ) } )