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 e. USGraph -> ( G e. ComplUSGraph <-> ( UnivVtx ` G ) = V ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iscusgrvtx.v | |- V = ( Vtx ` G ) |
|
| 2 | iscusgr | |- ( G e. ComplUSGraph <-> ( G e. USGraph /\ G e. ComplGraph ) ) |
|
| 3 | ibar | |- ( G e. USGraph -> ( G e. ComplGraph <-> ( G e. USGraph /\ G e. ComplGraph ) ) ) |
|
| 4 | 1 | cplgruvtxb | |- ( G e. USGraph -> ( G e. ComplGraph <-> ( UnivVtx ` G ) = V ) ) |
| 5 | 3 4 | bitr3d | |- ( G e. USGraph -> ( ( G e. USGraph /\ G e. ComplGraph ) <-> ( UnivVtx ` G ) = V ) ) |
| 6 | 2 5 | syl5bb | |- ( G e. USGraph -> ( G e. ComplUSGraph <-> ( UnivVtx ` G ) = V ) ) |