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

Metamath Proof Explorer

Theorem uvtxusgr

Description: The set of all universal vertices of a simple graph. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 31-Oct-2020)

Ref Expression
Hypotheses uvtxnbgr.v 
|- V = ( Vtx ` G )
uvtxusgr.e 
|- E = ( Edg ` G )
Assertion uvtxusgr 
|- ( G e. USGraph -> ( UnivVtx ` G ) = { n e. V | A. k e. ( V \ { n } ) { k , n } e. E } )

Proof

Step Hyp Ref Expression
1 uvtxnbgr.v 
 |-  V = ( Vtx ` G )
2 uvtxusgr.e 
 |-  E = ( Edg ` G )
3 1 uvtxval 
 |-  ( UnivVtx ` G ) = { n e. V | A. k e. ( V \ { n } ) k e. ( G NeighbVtx n ) }
4 2 nbusgreledg 
 |-  ( G e. USGraph -> ( k e. ( G NeighbVtx n ) <-> { k , n } e. E ) )
5 4 ralbidv 
 |-  ( G e. USGraph -> ( A. k e. ( V \ { n } ) k e. ( G NeighbVtx n ) <-> A. k e. ( V \ { n } ) { k , n } e. E ) )
6 5 rabbidv 
 |-  ( G e. USGraph -> { n e. V | A. k e. ( V \ { n } ) k e. ( G NeighbVtx n ) } = { n e. V | A. k e. ( V \ { n } ) { k , n } e. E } )
7 3 6 eqtrid 
 |-  ( G e. USGraph -> ( UnivVtx ` G ) = { n e. V | A. k e. ( V \ { n } ) { k , n } e. E } )