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Theorem nbgrssovtx 39596
Description: The neighbors of a vertex are a subset of all vertices except the vertex itself. Stronger version of nbgrssvtx 39592. (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 39591 . . . 4  |-  ( ( G  e.  W  /\  v  e.  ( G NeighbVtx  N ) )  ->  v  e.  V )
3 nbgrnself2 39595 . . . . . . . . . 10  |-  ( G  e.  W  ->  N  e/  ( G NeighbVtx  N )
)
43adantr 472 . . . . . . . . 9  |-  ( ( G  e.  W  /\  v  =  N )  ->  N  e/  ( G NeighbVtx  N ) )
5 df-nel 2644 . . . . . . . . . 10  |-  ( v  e/  ( G NeighbVtx  N )  <->  -.  v  e.  ( G NeighbVtx  N ) )
6 neleq1 2748 . . . . . . . . . . 11  |-  ( v  =  N  ->  (
v  e/  ( G NeighbVtx  N )  <->  N  e/  ( G NeighbVtx  N ) ) )
76adantl 473 . . . . . . . . . 10  |-  ( ( G  e.  W  /\  v  =  N )  ->  ( v  e/  ( G NeighbVtx  N )  <->  N  e/  ( G NeighbVtx  N ) ) )
85, 7syl5bbr 267 . . . . . . . . 9  |-  ( ( G  e.  W  /\  v  =  N )  ->  ( -.  v  e.  ( G NeighbVtx  N )  <->  N  e/  ( G NeighbVtx  N ) ) )
94, 8mpbird 240 . . . . . . . 8  |-  ( ( G  e.  W  /\  v  =  N )  ->  -.  v  e.  ( G NeighbVtx  N ) )
109ex 441 . . . . . . 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 436 . . . . 5  |-  ( ( G  e.  W  /\  v  e.  ( G NeighbVtx  N ) )  ->  -.  v  =  N )
1312neqned 2650 . . . 4  |-  ( ( G  e.  W  /\  v  e.  ( G NeighbVtx  N ) )  ->  v  =/=  N )
14 eldifsn 4088 . . . 4  |-  ( v  e.  ( V  \  { N } )  <->  ( v  e.  V  /\  v  =/=  N ) )
152, 13, 14sylanbrc 677 . . 3  |-  ( ( G  e.  W  /\  v  e.  ( G NeighbVtx  N ) )  ->  v  e.  ( V  \  { N } ) )
1615ex 441 . 2  |-  ( G  e.  W  ->  (
v  e.  ( G NeighbVtx  N )  ->  v  e.  ( V  \  { N } ) ) )
1716ssrdv 3424 1  |-  ( G  e.  W  ->  ( G NeighbVtx  N )  C_  ( V  \  { N }
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641    e/ wnel 2642    \ cdif 3387    C_ wss 3390   {csn 3959   ` cfv 5589  (class class class)co 6308  Vtxcvtx 39251   NeighbVtx cnbgr 39561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-nbgr 39565
This theorem is referenced by:  nbgrssvwo2  39597  usgrnbssovtx  39599  nbfusgrlevtxm1  39615  uvtxnbgr  39637  nbusgrvtxm1uvtx  39642  nbupgruvtxres  39644
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