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Theorem nbgra0nb 25236
Description: A vertex which is not endpoint of an edge has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
nbgra0nb  |-  ( V USGrph  E  ->  ( A. x  e.  ran  E  N  e/  x  ->  ( <. V ,  E >. Neighbors  N )  =  (/) ) )
Distinct variable groups:    x, V    x, E    x, N

Proof of Theorem nbgra0nb
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 nbusgra 25235 . . . 4  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } )
21adantr 472 . . 3  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
3 neleq2 2749 . . . . . . . . . 10  |-  ( x  =  { N ,  n }  ->  ( N  e/  x  <->  N  e/  { N ,  n }
) )
43rspcv 3132 . . . . . . . . 9  |-  ( { N ,  n }  e.  ran  E  ->  ( A. x  e.  ran  E  N  e/  x  ->  N  e/  { N ,  n } ) )
5 df-nel 2644 . . . . . . . . . 10  |-  ( N  e/  { N ,  n }  <->  -.  N  e.  { N ,  n }
)
6 elprg 3975 . . . . . . . . . . . . . 14  |-  ( N  e.  V  ->  ( N  e.  { N ,  n }  <->  ( N  =  N  \/  N  =  n ) ) )
76notbid 301 . . . . . . . . . . . . 13  |-  ( N  e.  V  ->  ( -.  N  e.  { N ,  n }  <->  -.  ( N  =  N  \/  N  =  n )
) )
8 ioran 498 . . . . . . . . . . . . 13  |-  ( -.  ( N  =  N  \/  N  =  n )  <->  ( -.  N  =  N  /\  -.  N  =  n ) )
97, 8syl6bb 269 . . . . . . . . . . . 12  |-  ( N  e.  V  ->  ( -.  N  e.  { N ,  n }  <->  ( -.  N  =  N  /\  -.  N  =  n
) ) )
10 eqid 2471 . . . . . . . . . . . . . 14  |-  N  =  N
1110pm2.24i 138 . . . . . . . . . . . . 13  |-  ( -.  N  =  N  -> 
( -.  N  =  n  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) ) )
1211imp 436 . . . . . . . . . . . 12  |-  ( ( -.  N  =  N  /\  -.  N  =  n )  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) )
139, 12syl6bi 236 . . . . . . . . . . 11  |-  ( N  e.  V  ->  ( -.  N  e.  { N ,  n }  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) ) )
14 usgraedgrnv 25183 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  { N ,  n }  e.  ran  E )  ->  ( N  e.  V  /\  n  e.  V ) )
1514simpld 466 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  { N ,  n }  e.  ran  E )  ->  N  e.  V )
1615ex 441 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( { N ,  n }  e.  ran  E  ->  N  e.  V
) )
1716con3d 140 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  ( -.  N  e.  V  ->  -.  { N ,  n }  e.  ran  E ) )
1817a1i 11 . . . . . . . . . . . . 13  |-  ( n  e.  V  ->  ( V USGrph  E  ->  ( -.  N  e.  V  ->  -. 
{ N ,  n }  e.  ran  E ) ) )
1918com13 82 . . . . . . . . . . . 12  |-  ( -.  N  e.  V  -> 
( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) )
2019a1d 25 . . . . . . . . . . 11  |-  ( -.  N  e.  V  -> 
( -.  N  e. 
{ N ,  n }  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) ) )
2113, 20pm2.61i 169 . . . . . . . . . 10  |-  ( -.  N  e.  { N ,  n }  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) )
225, 21sylbi 200 . . . . . . . . 9  |-  ( N  e/  { N ,  n }  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) )
234, 22syl6 33 . . . . . . . 8  |-  ( { N ,  n }  e.  ran  E  ->  ( A. x  e.  ran  E  N  e/  x  -> 
( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) ) )
2423com13 82 . . . . . . 7  |-  ( V USGrph  E  ->  ( A. x  e.  ran  E  N  e/  x  ->  ( { N ,  n }  e.  ran  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) ) )
2524imp 436 . . . . . 6  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  ( { N ,  n }  e.  ran  E  ->  (
n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) )
26 ax-1 6 . . . . . 6  |-  ( -. 
{ N ,  n }  e.  ran  E  -> 
( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) )
2725, 26pm2.61d1 164 . . . . 5  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  (
n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) )
2827ralrimiv 2808 . . . 4  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  A. n  e.  V  -.  { N ,  n }  e.  ran  E )
29 rabeq0 3757 . . . 4  |-  ( { n  e.  V  |  { N ,  n }  e.  ran  E }  =  (/)  <->  A. n  e.  V  -.  { N ,  n }  e.  ran  E )
3028, 29sylibr 217 . . 3  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  { n  e.  V  |  { N ,  n }  e.  ran  E }  =  (/) )
312, 30eqtrd 2505 . 2  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  ( <. V ,  E >. Neighbors  N
)  =  (/) )
3231ex 441 1  |-  ( V USGrph  E  ->  ( A. x  e.  ran  E  N  e/  x  ->  ( <. V ,  E >. Neighbors  N )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    e/ wnel 2642   A.wral 2756   {crab 2760   (/)c0 3722   {cpr 3961   <.cop 3965   class class class wbr 4395   ran crn 4840  (class class class)co 6308   USGrph cusg 25136   Neighbors cnbgra 25224
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  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  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-reu 2763  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-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-usgra 25139  df-nbgra 25227
This theorem is referenced by: (None)
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