Metamath Proof Explorer


Theorem iscplgrnb

Description: A graph is complete iff all vertices are neighbors of all vertices. (Contributed by AV, 1-Nov-2020)

Ref Expression
Hypothesis cplgruvtxb.v 𝑉 = ( Vtx ‘ 𝐺 )
Assertion iscplgrnb ( 𝐺𝑊 → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣𝑉𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) )

Proof

Step Hyp Ref Expression
1 cplgruvtxb.v 𝑉 = ( Vtx ‘ 𝐺 )
2 1 iscplgr ( 𝐺𝑊 → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
3 1 uvtxel ( 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝑣𝑉 ∧ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) )
4 3 a1i ( 𝐺𝑊 → ( 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝑣𝑉 ∧ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ) )
5 4 baibd ( ( 𝐺𝑊𝑣𝑉 ) → ( 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) )
6 5 ralbidva ( 𝐺𝑊 → ( ∀ 𝑣𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ∀ 𝑣𝑉𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) )
7 2 6 bitrd ( 𝐺𝑊 → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣𝑉𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) )