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Metamath Proof Explorer

Theorem uvtx2vtx1edgb

Description: If a hypergraph has two vertices, there is an edge between the vertices iff each vertex is universal. (Contributed by AV, 3-Nov-2020)

Ref Expression
Hypotheses uvtxel.v 
|- V = ( Vtx ` G )
isuvtx.e 
|- E = ( Edg ` G )
Assertion uvtx2vtx1edgb 
|- ( ( G e. UHGraph /\ ( # ` V ) = 2 ) -> ( V e. E <-> A. v e. V v e. ( UnivVtx ` G ) ) )

Proof

Step Hyp Ref Expression
1 uvtxel.v 
 |-  V = ( Vtx ` G )
2 isuvtx.e 
 |-  E = ( Edg ` G )
3 1 2 nbuhgr2vtx1edgb 
 |-  ( ( G e. UHGraph /\ ( # ` V ) = 2 ) -> ( V e. E <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )
4 1 uvtxel 
 |-  ( v e. ( UnivVtx ` G ) <-> ( v e. V /\ A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )
5 4 a1i 
 |-  ( ( G e. UHGraph /\ ( # ` V ) = 2 ) -> ( v e. ( UnivVtx ` G ) <-> ( v e. V /\ A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) ) )
6 5 baibd 
 |-  ( ( ( G e. UHGraph /\ ( # ` V ) = 2 ) /\ v e. V ) -> ( v e. ( UnivVtx ` G ) <-> A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )
7 6 bicomd 
 |-  ( ( ( G e. UHGraph /\ ( # ` V ) = 2 ) /\ v e. V ) -> ( A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) <-> v e. ( UnivVtx ` G ) ) )
8 7 ralbidva 
 |-  ( ( G e. UHGraph /\ ( # ` V ) = 2 ) -> ( A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) <-> A. v e. V v e. ( UnivVtx ` G ) ) )
9 3 8 bitrd 
 |-  ( ( G e. UHGraph /\ ( # ` V ) = 2 ) -> ( V e. E <-> A. v e. V v e. ( UnivVtx ` G ) ) )