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Theorem nbgra0nb 24728
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 24727 . . . 4  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } )
21adantr 463 . . 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 2741 . . . . . . . . . 10  |-  ( x  =  { N ,  n }  ->  ( N  e/  x  <->  N  e/  { N ,  n }
) )
43rspcv 3153 . . . . . . . . 9  |-  ( { N ,  n }  e.  ran  E  ->  ( A. x  e.  ran  E  N  e/  x  ->  N  e/  { N ,  n } ) )
5 df-nel 2599 . . . . . . . . . 10  |-  ( N  e/  { N ,  n }  <->  -.  N  e.  { N ,  n }
)
6 elprg 3985 . . . . . . . . . . . . . 14  |-  ( N  e.  V  ->  ( N  e.  { N ,  n }  <->  ( N  =  N  \/  N  =  n ) ) )
76notbid 292 . . . . . . . . . . . . 13  |-  ( N  e.  V  ->  ( -.  N  e.  { N ,  n }  <->  -.  ( N  =  N  \/  N  =  n )
) )
8 ioran 488 . . . . . . . . . . . . 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 2400 . . . . . . . . . . . . . 14  |-  N  =  N
1110pm2.24i 144 . . . . . . . . . . . . 13  |-  ( -.  N  =  N  -> 
( -.  N  =  n  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) ) )
1211imp 427 . . . . . . . . . . . 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 24676 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  { N ,  n }  e.  ran  E )  ->  ( N  e.  V  /\  n  e.  V ) )
1514simpld 457 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  { N ,  n }  e.  ran  E )  ->  N  e.  V )
1615ex 432 . . . . . . . . . . . . . . 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 31 . . . . . . . 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 427 . . . . . 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 2813 . . . 4  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  A. n  e.  V  -.  { N ,  n }  e.  ran  E )
29 rabeq0 3758 . . . 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 2441 . 2  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  ( <. V ,  E >. Neighbors  N
)  =  (/) )
3231ex 432 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 366    /\ wa 367    = wceq 1403    e. wcel 1840    e/ wnel 2597   A.wral 2751   {crab 2755   (/)c0 3735   {cpr 3971   <.cop 3975   class class class wbr 4392   ran crn 4941  (class class class)co 6232   USGrph cusg 24629   Neighbors cnbgra 24716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  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 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-card 8270  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-hash 12358  df-usgra 24632  df-nbgra 24719
This theorem is referenced by: (None)
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