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Metamath Proof Explorer

Theorem clnbgrcl

Description: If a class X has at least one element in its closed neighborhood, this class must be a vertex. (Contributed by AV, 7-May-2025)

Ref Expression
Hypothesis clnbgrcl.v 
|- V = ( Vtx ` G )
Assertion clnbgrcl 
|- ( N e. ( G ClNeighbVtx X ) -> X e. V )

Proof

Step Hyp Ref Expression
1 clnbgrcl.v 
 |-  V = ( Vtx ` G )
2 df-clnbgr 
 |-  ClNeighbVtx = ( g e. _V , v e. ( Vtx ` g ) |-> ( { v } u. { n e. ( Vtx ` g ) | E. e e. ( Edg ` g ) { v , n } C_ e } ) )
3 2 mpoxeldm 
 |-  ( N e. ( G ClNeighbVtx X ) -> ( G e. _V /\ X e. [_ G / g ]_ ( Vtx ` g ) ) )
4 csbfv 
 |-  [_ G / g ]_ ( Vtx ` g ) = ( Vtx ` G )
5 4 1 eqtr4i 
 |-  [_ G / g ]_ ( Vtx ` g ) = V
6 5 eleq2i 
 |-  ( X e. [_ G / g ]_ ( Vtx ` g ) <-> X e. V )
7 6 biimpi 
 |-  ( X e. [_ G / g ]_ ( Vtx ` g ) -> X e. V )
8 3 7 simpl2im 
 |-  ( N e. ( G ClNeighbVtx X ) -> X e. V )