Theorem nbcplgr

Description: In a complete graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 3-Nov-2020)

	Ref	Expression
Hypothesis	nbcplgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	nbcplgr	⊢ ( ( 𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉 ) → ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) )

Proof

Step	Hyp	Ref	Expression
1		nbcplgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	cplgruvtxb	⊢ ( 𝐺 ∈ ComplGraph → ( 𝐺 ∈ ComplGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) )
3	2	ibi	⊢ ( 𝐺 ∈ ComplGraph → ( UnivVtx ‘ 𝐺 ) = 𝑉 )
4	3	eqcomd	⊢ ( 𝐺 ∈ ComplGraph → 𝑉 = ( UnivVtx ‘ 𝐺 ) )
5	4	eleq2d	⊢ ( 𝐺 ∈ ComplGraph → ( 𝑁 ∈ 𝑉 ↔ 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ) )
6	5	biimpa	⊢ ( ( 𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉 ) → 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) )
7	1	uvtxnbgrb	⊢ ( 𝑁 ∈ 𝑉 → ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) ) )
8	7	adantl	⊢ ( ( 𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉 ) → ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) ) )
9	6 8	mpbid	⊢ ( ( 𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉 ) → ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) )