Description: A null graph (with no vertices) is a complete graph. (Contributed by Alexander van der Vekens, 13-Oct-2017) (Revised by AV, 1-Nov-2020)
Ref | Expression | ||
---|---|---|---|
Hypothesis | cplgr0v.v | ⊢ 𝑉 = ( Vtx ‘ 𝐺 ) | |
Assertion | cplgr0v | ⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ) → 𝐺 ∈ ComplGraph ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cplgr0v.v | ⊢ 𝑉 = ( Vtx ‘ 𝐺 ) | |
2 | rzal | ⊢ ( 𝑉 = ∅ → ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) | |
3 | 2 | adantl | ⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ) → ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) |
4 | 1 | iscplgr | ⊢ ( 𝐺 ∈ 𝑊 → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) ) |
5 | 4 | adantr | ⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ) → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) ) |
6 | 3 5 | mpbird | ⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ) → 𝐺 ∈ ComplGraph ) |