Theorem iscusgrvtx

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

	Ref	Expression
Hypothesis	iscusgrvtx.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	iscusgrvtx	⊢ ( 𝐺 ∈ ComplUSGraph ↔ ( 𝐺 ∈ USGraph ∧ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )

Proof

Step	Hyp	Ref	Expression
1		iscusgrvtx.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		iscusgr	⊢ ( 𝐺 ∈ ComplUSGraph ↔ ( 𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph ) )
3	1	iscplgr	⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
4	3	pm5.32i	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph ) ↔ ( 𝐺 ∈ USGraph ∧ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
5	2 4	bitri	⊢ ( 𝐺 ∈ ComplUSGraph ↔ ( 𝐺 ∈ USGraph ∧ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )