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 ) ) )