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Theorem nbgrssovtx 39432
Description: The neighbors of a vertex are a subset of all vertices except the vertex itself. Stronger version of nbgrssvtx 39428. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)
Hypothesis
Ref Expression
nbgrssovtx.v  |-  V  =  (Vtx `  G )
Assertion
Ref Expression
nbgrssovtx  |-  ( G  e.  W  ->  ( G NeighbVtx  N )  C_  ( V  \  { N }
) )

Proof of Theorem nbgrssovtx
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 nbgrssovtx.v . . . . 5  |-  V  =  (Vtx `  G )
21nbgrisvtx 39427 . . . 4  |-  ( ( G  e.  W  /\  v  e.  ( G NeighbVtx  N ) )  ->  v  e.  V )
3 nbgrnself2 39431 . . . . . . . . . 10  |-  ( G  e.  W  ->  N  e/  ( G NeighbVtx  N )
)
43adantr 467 . . . . . . . . 9  |-  ( ( G  e.  W  /\  v  =  N )  ->  N  e/  ( G NeighbVtx  N ) )
5 df-nel 2625 . . . . . . . . . 10  |-  ( v  e/  ( G NeighbVtx  N )  <->  -.  v  e.  ( G NeighbVtx  N ) )
6 neleq1 2729 . . . . . . . . . . 11  |-  ( v  =  N  ->  (
v  e/  ( G NeighbVtx  N )  <->  N  e/  ( G NeighbVtx  N ) ) )
76adantl 468 . . . . . . . . . 10  |-  ( ( G  e.  W  /\  v  =  N )  ->  ( v  e/  ( G NeighbVtx  N )  <->  N  e/  ( G NeighbVtx  N ) ) )
85, 7syl5bbr 263 . . . . . . . . 9  |-  ( ( G  e.  W  /\  v  =  N )  ->  ( -.  v  e.  ( G NeighbVtx  N )  <->  N  e/  ( G NeighbVtx  N ) ) )
94, 8mpbird 236 . . . . . . . 8  |-  ( ( G  e.  W  /\  v  =  N )  ->  -.  v  e.  ( G NeighbVtx  N ) )
109ex 436 . . . . . . 7  |-  ( G  e.  W  ->  (
v  =  N  ->  -.  v  e.  ( G NeighbVtx  N ) ) )
1110con2d 119 . . . . . 6  |-  ( G  e.  W  ->  (
v  e.  ( G NeighbVtx  N )  ->  -.  v  =  N )
)
1211imp 431 . . . . 5  |-  ( ( G  e.  W  /\  v  e.  ( G NeighbVtx  N ) )  ->  -.  v  =  N )
1312neqned 2631 . . . 4  |-  ( ( G  e.  W  /\  v  e.  ( G NeighbVtx  N ) )  ->  v  =/=  N )
14 eldifsn 4097 . . . 4  |-  ( v  e.  ( V  \  { N } )  <->  ( v  e.  V  /\  v  =/=  N ) )
152, 13, 14sylanbrc 670 . . 3  |-  ( ( G  e.  W  /\  v  e.  ( G NeighbVtx  N ) )  ->  v  e.  ( V  \  { N } ) )
1615ex 436 . 2  |-  ( G  e.  W  ->  (
v  e.  ( G NeighbVtx  N )  ->  v  e.  ( V  \  { N } ) ) )
1716ssrdv 3438 1  |-  ( G  e.  W  ->  ( G NeighbVtx  N )  C_  ( V  \  { N }
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622    e/ wnel 2623    \ cdif 3401    C_ wss 3404   {csn 3968   ` cfv 5582  (class class class)co 6290  Vtxcvtx 39101   NeighbVtx cnbgr 39397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-nbgr 39401
This theorem is referenced by:  nbgrssvwo2  39433  usgrnbssovtx  39435  nbfusgrlevtxm1  39451  uvtxnbgr  39473  nbusgrvtxm1uvtx  39478  nbupgruvtxres  39480
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