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 ) )