Description: A simple graph is complete iff the set of vertices is the set of universal vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017) (Revised by Alexander van der Vekens, 18-Jan-2018) (Revised by AV, 1-Nov-2020)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | iscusgrvtx.v | ⊢ 𝑉 = ( Vtx ‘ 𝐺 ) | |
| Assertion | cusgruvtxb | ⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplUSGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iscusgrvtx.v | ⊢ 𝑉 = ( Vtx ‘ 𝐺 ) | |
| 2 | iscusgr | ⊢ ( 𝐺 ∈ ComplUSGraph ↔ ( 𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph ) ) | |
| 3 | ibar | ⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplGraph ↔ ( 𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph ) ) ) | |
| 4 | 1 | cplgruvtxb | ⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) ) |
| 5 | 3 4 | bitr3d | ⊢ ( 𝐺 ∈ USGraph → ( ( 𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph ) ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) ) |
| 6 | 2 5 | syl5bb | ⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplUSGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) ) |