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Metamath Proof Explorer

Theorem iscplgrnb

Description: A graph is complete iff all vertices are neighbors of all vertices. (Contributed by AV, 1-Nov-2020)

Ref Expression
Hypothesis cplgruvtxb.v 
|- V = ( Vtx ` G )
Assertion iscplgrnb 
|- ( G e. W -> ( G e. ComplGraph <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )

Proof

Step Hyp Ref Expression
1 cplgruvtxb.v 
 |-  V = ( Vtx ` G )
2 1 iscplgr 
 |-  ( G e. W -> ( G e. ComplGraph <-> A. v e. V v e. ( UnivVtx ` G ) ) )
3 1 uvtxel 
 |-  ( v e. ( UnivVtx ` G ) <-> ( v e. V /\ A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )
4 3 a1i 
 |-  ( G e. W -> ( v e. ( UnivVtx ` G ) <-> ( v e. V /\ A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) ) )
5 4 baibd 
 |-  ( ( G e. W /\ v e. V ) -> ( v e. ( UnivVtx ` G ) <-> A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )
6 5 ralbidva 
 |-  ( G e. W -> ( A. v e. V v e. ( UnivVtx ` G ) <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )
7 2 6 bitrd 
 |-  ( G e. W -> ( G e. ComplGraph <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )