Metamath Proof Explorer


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