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Metamath Proof Explorer

Theorem iscusgredg

Description: A simple graph is complete iff all vertices are connected by an edge. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 1-Nov-2020)

Ref Expression
Hypotheses iscusgrvtx.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
iscusgredg.v ⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion iscusgredg ( 𝐺 ∈ ComplUSGraph ↔ ( 𝐺 ∈ USGraph ∧ ∀ 𝑘 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑘 } ) { 𝑛 , 𝑘 } ∈ 𝐸 ) )

Proof

Step Hyp Ref Expression
1 iscusgrvtx.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 iscusgredg.v ⊢ 𝐸 = ( Edg ‘ 𝐺 )
3 iscusgr ⊢ ( 𝐺 ∈ ComplUSGraph ↔ ( 𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph ) )
4 1 iscplgrnb ⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑘 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑘 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑘 ) ) )
5 2 nbusgreledg ⊢ ( 𝐺 ∈ USGraph → ( 𝑛 ∈ ( 𝐺 NeighbVtx 𝑘 ) ↔ { 𝑛 , 𝑘 } ∈ 𝐸 ) )
6 5 2ralbidv ⊢ ( 𝐺 ∈ USGraph → ( ∀ 𝑘 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑘 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑘 ) ↔ ∀ 𝑘 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑘 } ) { 𝑛 , 𝑘 } ∈ 𝐸 ) )
7 4 6 bitrd ⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑘 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑘 } ) { 𝑛 , 𝑘 } ∈ 𝐸 ) )
8 7 pm5.32i ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph ) ↔ ( 𝐺 ∈ USGraph ∧ ∀ 𝑘 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑘 } ) { 𝑛 , 𝑘 } ∈ 𝐸 ) )
9 3 8 bitri ⊢ ( 𝐺 ∈ ComplUSGraph ↔ ( 𝐺 ∈ USGraph ∧ ∀ 𝑘 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑘 } ) { 𝑛 , 𝑘 } ∈ 𝐸 ) )