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Metamath Proof Explorer

Theorem uvtxel

Description: A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 29-Oct-2020) (Revised by AV, 14-Feb-2022)

Ref Expression
Hypothesis uvtxel.v 
|- V = ( Vtx ` G )
Assertion uvtxel 
|- ( N e. ( UnivVtx ` G ) <-> ( N e. V /\ A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) ) )

Proof

Step Hyp Ref Expression
1 uvtxel.v 
 |-  V = ( Vtx ` G )
2 sneq 
 |-  ( v = N -> { v } = { N } )
3 2 difeq2d 
 |-  ( v = N -> ( V \ { v } ) = ( V \ { N } ) )
4 oveq2 
 |-  ( v = N -> ( G NeighbVtx v ) = ( G NeighbVtx N ) )
5 4 eleq2d 
 |-  ( v = N -> ( n e. ( G NeighbVtx v ) <-> n e. ( G NeighbVtx N ) ) )
6 3 5 raleqbidv 
 |-  ( v = N -> ( A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) <-> A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) ) )
7 1 uvtxval 
 |-  ( UnivVtx ` G ) = { v e. V | A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) }
8 6 7 elrab2 
 |-  ( N e. ( UnivVtx ` G ) <-> ( N e. V /\ A. n e. ( V \ { N } ) n e. ( G NeighbVtx N ) ) )