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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 ( 𝐺 ∈ 𝑊 → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) )

Proof

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