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Theorem uvtxnbgra 24910
Description: A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
Assertion
Ref Expression
uvtxnbgra  |-  ( ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
)  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } ) )

Proof of Theorem uvtxnbgra
Dummy variables  v  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 24755 . . . 4  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 uvtxel 24906 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( N  e.  ( V UnivVertex  E )  <->  ( N  e.  V  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E ) ) )
31, 2syl 17 . . 3  |-  ( V USGrph  E  ->  ( N  e.  ( V UnivVertex  E )  <->  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) ) )
4 nbusgra 24845 . . . . . 6  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } )
54adantr 463 . . . . 5  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
6 preq2 4052 . . . . . . . . . 10  |-  ( n  =  x  ->  { N ,  n }  =  { N ,  x }
)
76eleq1d 2471 . . . . . . . . 9  |-  ( n  =  x  ->  ( { N ,  n }  e.  ran  E  <->  { N ,  x }  e.  ran  E ) )
87elrab 3207 . . . . . . . 8  |-  ( x  e.  { n  e.  V  |  { N ,  n }  e.  ran  E }  <->  ( x  e.  V  /\  { N ,  x }  e.  ran  E ) )
9 usgraedgrn 24798 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  { N ,  x }  e.  ran  E )  ->  N  =/=  x )
10 simpr 459 . . . . . . . . . . . . . . . 16  |-  ( ( N  =/=  x  /\  x  e.  V )  ->  x  e.  V )
11 elsni 3997 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  { N }  ->  x  =  N )
1211eqcomd 2410 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  { N }  ->  N  =  x )
1312necon3ai 2631 . . . . . . . . . . . . . . . . 17  |-  ( N  =/=  x  ->  -.  x  e.  { N } )
1413adantr 463 . . . . . . . . . . . . . . . 16  |-  ( ( N  =/=  x  /\  x  e.  V )  ->  -.  x  e.  { N } )
1510, 14eldifd 3425 . . . . . . . . . . . . . . 15  |-  ( ( N  =/=  x  /\  x  e.  V )  ->  x  e.  ( V 
\  { N }
) )
1615ex 432 . . . . . . . . . . . . . 14  |-  ( N  =/=  x  ->  (
x  e.  V  ->  x  e.  ( V  \  { N } ) ) )
179, 16syl 17 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  { N ,  x }  e.  ran  E )  ->  ( x  e.  V  ->  x  e.  ( V  \  { N } ) ) )
1817ex 432 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( { N ,  x }  e.  ran  E  ->  ( x  e.  V  ->  x  e.  ( V  \  { N } ) ) ) )
1918com13 80 . . . . . . . . . . 11  |-  ( x  e.  V  ->  ( { N ,  x }  e.  ran  E  ->  ( V USGrph  E  ->  x  e.  ( V  \  { N } ) ) ) )
2019imp 427 . . . . . . . . . 10  |-  ( ( x  e.  V  /\  { N ,  x }  e.  ran  E )  -> 
( V USGrph  E  ->  x  e.  ( V  \  { N } ) ) )
2120com12 29 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( x  e.  V  /\  { N ,  x }  e.  ran  E )  ->  x  e.  ( V  \  { N } ) ) )
2221adantr 463 . . . . . . . 8  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  (
( x  e.  V  /\  { N ,  x }  e.  ran  E )  ->  x  e.  ( V  \  { N } ) ) )
238, 22syl5bi 217 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  (
x  e.  { n  e.  V  |  { N ,  n }  e.  ran  E }  ->  x  e.  ( V  \  { N } ) ) )
24 eldifi 3565 . . . . . . . . . 10  |-  ( x  e.  ( V  \  { N } )  ->  x  e.  V )
2524adantl 464 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E ) )  /\  x  e.  ( V  \  { N } ) )  ->  x  e.  V )
26 preq1 4051 . . . . . . . . . . . . . . . . 17  |-  ( v  =  x  ->  { v ,  N }  =  { x ,  N } )
2726eleq1d 2471 . . . . . . . . . . . . . . . 16  |-  ( v  =  x  ->  ( { v ,  N }  e.  ran  E  <->  { x ,  N }  e.  ran  E ) )
2827rspcva 3158 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( V 
\  { N }
)  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E )  ->  { x ,  N }  e.  ran  E )
29 prcom 4050 . . . . . . . . . . . . . . . . . . 19  |-  { x ,  N }  =  { N ,  x }
3029eleq1i 2479 . . . . . . . . . . . . . . . . . 18  |-  ( { x ,  N }  e.  ran  E  <->  { N ,  x }  e.  ran  E )
3130biimpi 194 . . . . . . . . . . . . . . . . 17  |-  ( { x ,  N }  e.  ran  E  ->  { N ,  x }  e.  ran  E )
3231a1d 25 . . . . . . . . . . . . . . . 16  |-  ( { x ,  N }  e.  ran  E  ->  ( N  e.  V  ->  { N ,  x }  e.  ran  E ) )
3332a1d 25 . . . . . . . . . . . . . . 15  |-  ( { x ,  N }  e.  ran  E  ->  ( V USGrph  E  ->  ( N  e.  V  ->  { N ,  x }  e.  ran  E ) ) )
3428, 33syl 17 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ( V 
\  { N }
)  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E )  ->  ( V USGrph  E  ->  ( N  e.  V  ->  { N ,  x }  e.  ran  E ) ) )
3534ex 432 . . . . . . . . . . . . 13  |-  ( x  e.  ( V  \  { N } )  -> 
( A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E  ->  ( V USGrph  E  ->  ( N  e.  V  ->  { N ,  x }  e.  ran  E ) ) ) )
3635com14 88 . . . . . . . . . . . 12  |-  ( N  e.  V  ->  ( A. v  e.  ( V  \  { N }
) { v ,  N }  e.  ran  E  ->  ( V USGrph  E  ->  ( x  e.  ( V  \  { N } )  ->  { N ,  x }  e.  ran  E ) ) ) )
3736imp 427 . . . . . . . . . . 11  |-  ( ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E )  ->  ( V USGrph  E  ->  ( x  e.  ( V  \  { N } )  ->  { N ,  x }  e.  ran  E ) ) )
3837impcom 428 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  (
x  e.  ( V 
\  { N }
)  ->  { N ,  x }  e.  ran  E ) )
3938imp 427 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E ) )  /\  x  e.  ( V  \  { N } ) )  ->  { N ,  x }  e.  ran  E )
4025, 39, 8sylanbrc 662 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E ) )  /\  x  e.  ( V  \  { N } ) )  ->  x  e.  { n  e.  V  |  { N ,  n }  e.  ran  E } )
4140ex 432 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  (
x  e.  ( V 
\  { N }
)  ->  x  e.  { n  e.  V  |  { N ,  n }  e.  ran  E } ) )
4223, 41impbid 190 . . . . . 6  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  (
x  e.  { n  e.  V  |  { N ,  n }  e.  ran  E }  <->  x  e.  ( V  \  { N } ) ) )
4342eqrdv 2399 . . . . 5  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  { n  e.  V  |  { N ,  n }  e.  ran  E }  =  ( V  \  { N } ) )
445, 43eqtrd 2443 . . . 4  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  ( <. V ,  E >. Neighbors  N
)  =  ( V 
\  { N }
) )
4544ex 432 . . 3  |-  ( V USGrph  E  ->  ( ( N  e.  V  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E )  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } ) ) )
463, 45sylbid 215 . 2  |-  ( V USGrph  E  ->  ( N  e.  ( V UnivVertex  E )  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } ) ) )
4746imp 427 1  |-  ( ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
)  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   {crab 2758   _Vcvv 3059    \ cdif 3411   {csn 3972   {cpr 3974   <.cop 3978   class class class wbr 4395   ran crn 4824  (class class class)co 6278   USGrph cusg 24747   Neighbors cnbgra 24834   UnivVertex cuvtx 24836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-hash 12453  df-usgra 24750  df-nbgra 24837  df-uvtx 24839
This theorem is referenced by:  uvtxnm1nbgra  24911  uvtxnbgravtx  24912  uvtxnb  24914  usgrauvtxvd  37987
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