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Theorem nbgranv0 23475
Description: There are no neighbors of a class which is not a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
nbgranv0  |-  ( N  e/  V  ->  ( <. V ,  E >. Neighbors  N
)  =  (/) )

Proof of Theorem nbgranv0
Dummy variables  g 
k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgra 23469 . 2  |- Neighbors  =  ( g  e.  _V , 
k  e.  ( 1st `  g )  |->  { n  e.  ( 1st `  g
)  |  { k ,  n }  e.  ran  ( 2nd `  g
) } )
21mpt2xopynvov0 6837 1  |-  ( N  e/  V  ->  ( <. V ,  E >. Neighbors  N
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    e/ wnel 2645   {crab 2799   (/)c0 3737   {cpr 3979   <.cop 3983   ran crn 4941   ` cfv 5518  (class class class)co 6192   1stc1st 6677   2ndc2nd 6678   Neighbors cnbgra 23466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-nbgra 23469
This theorem is referenced by:  nbusgra  23476  nbgranself2  23484
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