MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nbgra0nb Structured version   Visualization version   Unicode version

Theorem nbgra0nb 25169
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 25168 . . . 4  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } )
21adantr 467 . . 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 2732 . . . . . . . . . 10  |-  ( x  =  { N ,  n }  ->  ( N  e/  x  <->  N  e/  { N ,  n }
) )
43rspcv 3148 . . . . . . . . 9  |-  ( { N ,  n }  e.  ran  E  ->  ( A. x  e.  ran  E  N  e/  x  ->  N  e/  { N ,  n } ) )
5 df-nel 2627 . . . . . . . . . 10  |-  ( N  e/  { N ,  n }  <->  -.  N  e.  { N ,  n }
)
6 elprg 3986 . . . . . . . . . . . . . 14  |-  ( N  e.  V  ->  ( N  e.  { N ,  n }  <->  ( N  =  N  \/  N  =  n ) ) )
76notbid 296 . . . . . . . . . . . . 13  |-  ( N  e.  V  ->  ( -.  N  e.  { N ,  n }  <->  -.  ( N  =  N  \/  N  =  n )
) )
8 ioran 493 . . . . . . . . . . . . 13  |-  ( -.  ( N  =  N  \/  N  =  n )  <->  ( -.  N  =  N  /\  -.  N  =  n ) )
97, 8syl6bb 265 . . . . . . . . . . . 12  |-  ( N  e.  V  ->  ( -.  N  e.  { N ,  n }  <->  ( -.  N  =  N  /\  -.  N  =  n
) ) )
10 eqid 2453 . . . . . . . . . . . . . 14  |-  N  =  N
1110pm2.24i 137 . . . . . . . . . . . . 13  |-  ( -.  N  =  N  -> 
( -.  N  =  n  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) ) )
1211imp 431 . . . . . . . . . . . 12  |-  ( ( -.  N  =  N  /\  -.  N  =  n )  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) )
139, 12syl6bi 232 . . . . . . . . . . 11  |-  ( N  e.  V  ->  ( -.  N  e.  { N ,  n }  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) ) )
14 usgraedgrnv 25116 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  { N ,  n }  e.  ran  E )  ->  ( N  e.  V  /\  n  e.  V ) )
1514simpld 461 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  { N ,  n }  e.  ran  E )  ->  N  e.  V )
1615ex 436 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( { N ,  n }  e.  ran  E  ->  N  e.  V
) )
1716con3d 139 . . . . . . . . . . . . . 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 83 . . . . . . . . . . . 12  |-  ( -.  N  e.  V  -> 
( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) )
2019a1d 26 . . . . . . . . . . 11  |-  ( -.  N  e.  V  -> 
( -.  N  e. 
{ N ,  n }  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) ) )
2113, 20pm2.61i 168 . . . . . . . . . 10  |-  ( -.  N  e.  { N ,  n }  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) )
225, 21sylbi 199 . . . . . . . . 9  |-  ( N  e/  { N ,  n }  ->  ( V USGrph  E  ->  ( n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) ) )
234, 22syl6 34 . . . . . . . 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 83 . . . . . . 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 431 . . . . . 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 163 . . . . 5  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  (
n  e.  V  ->  -.  { N ,  n }  e.  ran  E ) )
2827ralrimiv 2802 . . . 4  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  A. n  e.  V  -.  { N ,  n }  e.  ran  E )
29 rabeq0 3756 . . . 4  |-  ( { n  e.  V  |  { N ,  n }  e.  ran  E }  =  (/)  <->  A. n  e.  V  -.  { N ,  n }  e.  ran  E )
3028, 29sylibr 216 . . 3  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  { n  e.  V  |  { N ,  n }  e.  ran  E }  =  (/) )
312, 30eqtrd 2487 . 2  |-  ( ( V USGrph  E  /\  A. x  e.  ran  E  N  e/  x )  ->  ( <. V ,  E >. Neighbors  N
)  =  (/) )
3231ex 436 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 370    /\ wa 371    = wceq 1446    e. wcel 1889    e/ wnel 2625   A.wral 2739   {crab 2743   (/)c0 3733   {cpr 3972   <.cop 3976   class class class wbr 4405   ran crn 4838  (class class class)co 6295   USGrph cusg 25069   Neighbors cnbgra 25157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12523  df-usgra 25072  df-nbgra 25160
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator