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Metamath Proof Explorer

Theorem elclnbgrelnbgr

Description: An element of the closed neighborhood of a vertex which is not the vertex itself is an element of the open neighborhood of the vertex. (Contributed by AV, 24-Sep-2025)

Ref Expression
Assertion elclnbgrelnbgr 
|- ( ( X e. ( G ClNeighbVtx N ) /\ X =/= N ) -> X e. ( G NeighbVtx N ) )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
2 1 clnbgrcl 
 |-  ( X e. ( G ClNeighbVtx N ) -> N e. ( Vtx ` G ) )
3 1 dfclnbgr4 
 |-  ( N e. ( Vtx ` G ) -> ( G ClNeighbVtx N ) = ( { N } u. ( G NeighbVtx N ) ) )
4 2 3 syl 
 |-  ( X e. ( G ClNeighbVtx N ) -> ( G ClNeighbVtx N ) = ( { N } u. ( G NeighbVtx N ) ) )
5 4 eleq2d 
 |-  ( X e. ( G ClNeighbVtx N ) -> ( X e. ( G ClNeighbVtx N ) <-> X e. ( { N } u. ( G NeighbVtx N ) ) ) )
6 elun 
 |-  ( X e. ( { N } u. ( G NeighbVtx N ) ) <-> ( X e. { N } \/ X e. ( G NeighbVtx N ) ) )
7 elsng 
 |-  ( X e. ( G ClNeighbVtx N ) -> ( X e. { N } <-> X = N ) )
8 eqneqall 
 |-  ( X = N -> ( X =/= N -> X e. ( G NeighbVtx N ) ) )
9 8 a1i 
 |-  ( X e. ( G ClNeighbVtx N ) -> ( X = N -> ( X =/= N -> X e. ( G NeighbVtx N ) ) ) )
10 7 9 sylbid 
 |-  ( X e. ( G ClNeighbVtx N ) -> ( X e. { N } -> ( X =/= N -> X e. ( G NeighbVtx N ) ) ) )
11 ax-1 
 |-  ( X e. ( G NeighbVtx N ) -> ( X =/= N -> X e. ( G NeighbVtx N ) ) )
12 11 a1i 
 |-  ( X e. ( G ClNeighbVtx N ) -> ( X e. ( G NeighbVtx N ) -> ( X =/= N -> X e. ( G NeighbVtx N ) ) ) )
13 10 12 jaod 
 |-  ( X e. ( G ClNeighbVtx N ) -> ( ( X e. { N } \/ X e. ( G NeighbVtx N ) ) -> ( X =/= N -> X e. ( G NeighbVtx N ) ) ) )
14 6 13 biimtrid 
 |-  ( X e. ( G ClNeighbVtx N ) -> ( X e. ( { N } u. ( G NeighbVtx N ) ) -> ( X =/= N -> X e. ( G NeighbVtx N ) ) ) )
15 5 14 sylbid 
 |-  ( X e. ( G ClNeighbVtx N ) -> ( X e. ( G ClNeighbVtx N ) -> ( X =/= N -> X e. ( G NeighbVtx N ) ) ) )
16 15 pm2.43i 
 |-  ( X e. ( G ClNeighbVtx N ) -> ( X =/= N -> X e. ( G NeighbVtx N ) ) )
17 16 imp 
 |-  ( ( X e. ( G ClNeighbVtx N ) /\ X =/= N ) -> X e. ( G NeighbVtx N ) )