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Theorem nbgra0nb 24093
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 24092 . . . 4  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } )
21adantr 465 . . 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 2802 . . . . . . . . . 10  |-  ( x  =  { N ,  n }  ->  ( N  e/  x  <->  N  e/  { N ,  n }
) )
43rspcv 3205 . . . . . . . . 9  |-  ( { N ,  n }  e.  ran  E  ->  ( A. x  e.  ran  E  N  e/  x  ->  N  e/  { N ,  n } ) )
5 df-nel 2660 . . . . . . . . . 10  |-  ( N  e/  { N ,  n }  <->  -.  N  e.  { N ,  n }
)
6 elprg 4038 . . . . . . . . . . . . . 14  |-  ( N  e.  V  ->  ( N  e.  { N ,  n }  <->  ( N  =  N  \/  N  =  n ) ) )
76notbid 294 . . . . . . . . . . . . 13  |-  ( N  e.  V  ->  ( -.  N  e.  { N ,  n }  <->  -.  ( N  =  N  \/  N  =  n )
) )
8 ioran 490 . . . . . . . . . . . . 13  |-  ( -.  ( N  =  N  \/  N  =  n )  <->  ( -.  N  =  N  /\  -.  N  =  n ) )
97, 8syl6bb 261 . . . . . . . . . . . 12  |-  ( N  e.  V  ->  ( -.  N  e.  { N ,  n }  <->  ( -.  N  =  N  /\  -.  N  =  n
) ) )
10 eqid 2462 . . . . . . . . . . . . . 14  |-  N  =  N
1110pm2.24i 144 . . . . . . . . . . . . 13  |-  ( -.  N  =  N  -> 
( -.  N  =  n  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) ) )
1211imp 429 . . . . . . . . . . . 12  |-  ( ( -.  N  =  N  /\  -.  N  =  n )  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) )
139, 12syl6bi 228 . . . . . . . . . . 11  |-  ( N  e.  V  ->  ( -.  N  e.  { N ,  n }  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) ) )
14 usgraedgrnv 24041 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  { N ,  n }  e.  ran  E )  ->  ( N  e.  V  /\  n  e.  V ) )
1514simpld 459 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  { N ,  n }  e.  ran  E )  ->  N  e.  V )
1615ex 434 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( { N ,  n }  e.  ran  E  ->  N  e.  V
) )
1716con3d 133 . . . . . . . . . . . . . 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 80 . . . . . . . . . . . 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 164 . . . . . . . . . 10  |-  ( -.  N  e.  { N ,  n }  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) )
225, 21sylbi 195 . . . . . . . . 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 80 . . . . . . 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 429 . . . . . 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 159 . . . . 5  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  (
n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) )
2827ralrimiv 2871 . . . 4  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  A. n  e.  V  -.  { N ,  n }  e.  ran  E )
29 rabeq0 3802 . . . 4  |-  ( { n  e.  V  |  { N ,  n }  e.  ran  E }  =  (/)  <->  A. n  e.  V  -.  { N ,  n }  e.  ran  E )
3028, 29sylibr 212 . . 3  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  { n  e.  V  |  { N ,  n }  e.  ran  E }  =  (/) )
312, 30eqtrd 2503 . 2  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  ( <. V ,  E >. Neighbors  N
)  =  (/) )
3231ex 434 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 368    /\ wa 369    = wceq 1374    e. wcel 1762    e/ wnel 2658   A.wral 2809   {crab 2813   (/)c0 3780   {cpr 4024   <.cop 4028   class class class wbr 4442   ran crn 4995  (class class class)co 6277   USGrph cusg 23995   Neighbors cnbgra 24081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-hash 12363  df-usgra 23998  df-nbgra 24084
This theorem is referenced by: (None)
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