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Theorem nbgrassovt 24562
Description: The neighbors of a vertex are a subset of the other vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
nbgrassovt  |-  ( V USGrph  E  ->  ( N  e.  V  ->  ( <. V ,  E >. Neighbors  N ) 
C_  ( V  \  { N } ) ) )

Proof of Theorem nbgrassovt
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 nbgranself 24561 . . . 4  |-  ( V USGrph  E  ->  A. n  e.  V  n  e/  ( <. V ,  E >. Neighbors  n ) )
2 id 22 . . . . . 6  |-  ( n  =  N  ->  n  =  N )
3 oveq2 6304 . . . . . 6  |-  ( n  =  N  ->  ( <. V ,  E >. Neighbors  n
)  =  ( <. V ,  E >. Neighbors  N
) )
42, 3neleq12d 2794 . . . . 5  |-  ( n  =  N  ->  (
n  e/  ( <. V ,  E >. Neighbors  n )  <-> 
N  e/  ( <. V ,  E >. Neighbors  N ) ) )
54rspcv 3206 . . . 4  |-  ( N  e.  V  ->  ( A. n  e.  V  n  e/  ( <. V ,  E >. Neighbors  n )  ->  N  e/  ( <. V ,  E >. Neighbors  N ) ) )
61, 5mpan9 469 . . 3  |-  ( ( V USGrph  E  /\  N  e.  V )  ->  N  e/  ( <. V ,  E >. Neighbors  N ) )
7 nbgrassvt 24560 . . . 4  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  C_  V
)
87adantr 465 . . 3  |-  ( ( V USGrph  E  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  C_  V )
9 df-nel 2655 . . . 4  |-  ( N  e/  ( <. V ,  E >. Neighbors  N )  <->  -.  N  e.  ( <. V ,  E >. Neighbors  N ) )
10 difsn 4166 . . . . 5  |-  ( -.  N  e.  ( <. V ,  E >. Neighbors  N
)  ->  ( ( <. V ,  E >. Neighbors  N
)  \  { N } )  =  (
<. V ,  E >. Neighbors  N
) )
11 simpl 457 . . . . . . 7  |-  ( ( ( ( <. V ,  E >. Neighbors  N )  \  { N } )  =  (
<. V ,  E >. Neighbors  N
)  /\  ( <. V ,  E >. Neighbors  N ) 
C_  V )  -> 
( ( <. V ,  E >. Neighbors  N )  \  { N } )  =  (
<. V ,  E >. Neighbors  N
) )
12 simpr 461 . . . . . . . 8  |-  ( ( ( ( <. V ,  E >. Neighbors  N )  \  { N } )  =  (
<. V ,  E >. Neighbors  N
)  /\  ( <. V ,  E >. Neighbors  N ) 
C_  V )  -> 
( <. V ,  E >. Neighbors  N )  C_  V
)
1312ssdifd 3636 . . . . . . 7  |-  ( ( ( ( <. V ,  E >. Neighbors  N )  \  { N } )  =  (
<. V ,  E >. Neighbors  N
)  /\  ( <. V ,  E >. Neighbors  N ) 
C_  V )  -> 
( ( <. V ,  E >. Neighbors  N )  \  { N } )  C_  ( V  \  { N }
) )
1411, 13eqsstr3d 3534 . . . . . 6  |-  ( ( ( ( <. V ,  E >. Neighbors  N )  \  { N } )  =  (
<. V ,  E >. Neighbors  N
)  /\  ( <. V ,  E >. Neighbors  N ) 
C_  V )  -> 
( <. V ,  E >. Neighbors  N )  C_  ( V  \  { N }
) )
1514ex 434 . . . . 5  |-  ( ( ( <. V ,  E >. Neighbors  N )  \  { N } )  =  (
<. V ,  E >. Neighbors  N
)  ->  ( ( <. V ,  E >. Neighbors  N
)  C_  V  ->  (
<. V ,  E >. Neighbors  N
)  C_  ( V  \  { N } ) ) )
1610, 15syl 16 . . . 4  |-  ( -.  N  e.  ( <. V ,  E >. Neighbors  N
)  ->  ( ( <. V ,  E >. Neighbors  N
)  C_  V  ->  (
<. V ,  E >. Neighbors  N
)  C_  ( V  \  { N } ) ) )
179, 16sylbi 195 . . 3  |-  ( N  e/  ( <. V ,  E >. Neighbors  N )  ->  (
( <. V ,  E >. Neighbors  N )  C_  V  ->  ( <. V ,  E >. Neighbors  N )  C_  ( V  \  { N }
) ) )
186, 8, 17sylc 60 . 2  |-  ( ( V USGrph  E  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  C_  ( V  \  { N } ) )
1918ex 434 1  |-  ( V USGrph  E  ->  ( N  e.  V  ->  ( <. V ,  E >. Neighbors  N ) 
C_  ( V  \  { N } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    e/ wnel 2653   A.wral 2807    \ cdif 3468    C_ wss 3471   {csn 4032   <.cop 4038   class class class wbr 4456  (class class class)co 6296   USGrph cusg 24457   Neighbors cnbgra 24544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409  df-usgra 24460  df-nbgra 24547
This theorem is referenced by:  nbgranself2  24563
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