Theorem cplgr2v

Description: An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by AV, 3-Nov-2020)

	Ref	Expression
Hypotheses	cplgr0v.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	cplgr2v.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	cplgr2v	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝐺 ∈ ComplGraph ↔ 𝑉 ∈ 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		cplgr0v.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		cplgr2v.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3	1	iscplgr	⊢ ( 𝐺 ∈ UHGraph → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
4	3	adantr	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
5	1 2	uvtx2vtx1edgb	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝑉 ∈ 𝐸 ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
6	4 5	bitr4d	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝐺 ∈ ComplGraph ↔ 𝑉 ∈ 𝐸 ) )