This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem uvtx2vtx1edg

Description: If a graph has two vertices, and there is an edge between the vertices, then each vertex is universal. (Contributed by AV, 1-Nov-2020) (Revised by AV, 25-Mar-2021) (Proof shortened by AV, 14-Feb-2022)

Ref Expression
Hypotheses uvtxel.v 
|- V = ( Vtx ` G )
isuvtx.e 
|- E = ( Edg ` G )
Assertion uvtx2vtx1edg 
|- ( ( ( # ` 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 nbgr2vtx1edg 
 |-  ( ( ( # ` 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 
 |-  ( ( ( # ` V ) = 2 /\ V e. E ) -> ( v e. ( UnivVtx ` G ) <-> ( v e. V /\ A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) ) )
6 5 baibd 
 |-  ( ( ( ( # ` V ) = 2 /\ V e. E ) /\ v e. V ) -> ( v e. ( UnivVtx ` G ) <-> A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )
7 6 ralbidva 
 |-  ( ( ( # ` V ) = 2 /\ V e. E ) -> ( A. v e. V v e. ( UnivVtx ` G ) <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )
8 3 7 mpbird 
 |-  ( ( ( # ` V ) = 2 /\ V e. E ) -> A. v e. V v e. ( UnivVtx ` G ) )