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Metamath Proof Explorer

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 } ) )