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Theorem uvtxnbgra 24620
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 24465 . . . 4  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 uvtxel 24616 . . . 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 16 . . 3  |-  ( V USGrph  E  ->  ( N  e.  ( V UnivVertex  E )  <->  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) ) )
4 nbusgra 24555 . . . . . 6  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } )
54adantr 465 . . . . 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 4112 . . . . . . . . . 10  |-  ( n  =  x  ->  { N ,  n }  =  { N ,  x }
)
76eleq1d 2526 . . . . . . . . 9  |-  ( n  =  x  ->  ( { N ,  n }  e.  ran  E  <->  { N ,  x }  e.  ran  E ) )
87elrab 3257 . . . . . . . 8  |-  ( x  e.  { n  e.  V  |  { N ,  n }  e.  ran  E }  <->  ( x  e.  V  /\  { N ,  x }  e.  ran  E ) )
9 usgraedgrn 24508 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  { N ,  x }  e.  ran  E )  ->  N  =/=  x )
10 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( N  =/=  x  /\  x  e.  V )  ->  x  e.  V )
11 elsni 4057 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  { N }  ->  x  =  N )
1211eqcomd 2465 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  { N }  ->  N  =  x )
1312necon3ai 2685 . . . . . . . . . . . . . . . . 17  |-  ( N  =/=  x  ->  -.  x  e.  { N } )
1413adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( N  =/=  x  /\  x  e.  V )  ->  -.  x  e.  { N } )
1510, 14eldifd 3482 . . . . . . . . . . . . . . 15  |-  ( ( N  =/=  x  /\  x  e.  V )  ->  x  e.  ( V 
\  { N }
) )
1615ex 434 . . . . . . . . . . . . . 14  |-  ( N  =/=  x  ->  (
x  e.  V  ->  x  e.  ( V  \  { N } ) ) )
179, 16syl 16 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  { N ,  x }  e.  ran  E )  ->  ( x  e.  V  ->  x  e.  ( V  \  { N } ) ) )
1817ex 434 . . . . . . . . . . . 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 429 . . . . . . . . . 10  |-  ( ( x  e.  V  /\  { N ,  x }  e.  ran  E )  -> 
( V USGrph  E  ->  x  e.  ( V  \  { N } ) ) )
2120com12 31 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( x  e.  V  /\  { N ,  x }  e.  ran  E )  ->  x  e.  ( V  \  { N } ) ) )
2221adantr 465 . . . . . . . 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 3622 . . . . . . . . . 10  |-  ( x  e.  ( V  \  { N } )  ->  x  e.  V )
2524adantl 466 . . . . . . . . 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 4111 . . . . . . . . . . . . . . . . 17  |-  ( v  =  x  ->  { v ,  N }  =  { x ,  N } )
2726eleq1d 2526 . . . . . . . . . . . . . . . 16  |-  ( v  =  x  ->  ( { v ,  N }  e.  ran  E  <->  { x ,  N }  e.  ran  E ) )
2827rspcva 3208 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( V 
\  { N }
)  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E )  ->  { x ,  N }  e.  ran  E )
29 prcom 4110 . . . . . . . . . . . . . . . . . . 19  |-  { x ,  N }  =  { N ,  x }
3029eleq1i 2534 . . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . 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 429 . . . . . . . . . . 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 430 . . . . . . . . . 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 429 . . . . . . . . 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 664 . . . . . . . 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 434 . . . . . . 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 191 . . . . . 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 2454 . . . . 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 2498 . . . 4  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  ( <. V ,  E >. Neighbors  N
)  =  ( V 
\  { N }
) )
4544ex 434 . . 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 429 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 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109    \ cdif 3468   {csn 4032   {cpr 4034   <.cop 4038   class class class wbr 4456   ran crn 5009  (class class class)co 6296   USGrph cusg 24457   Neighbors cnbgra 24544   UnivVertex cuvtx 24546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409  df-usgra 24460  df-nbgra 24547  df-uvtx 24549
This theorem is referenced by:  uvtxnm1nbgra  24621  uvtxnbgravtx  24622  uvtxnb  24624  usgrauvtxvd  32620
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