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Theorem uvtxnbgravtx 23540
Description: A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
Assertion
Ref Expression
uvtxnbgravtx  |-  ( ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
)  ->  A. v  e.  ( V  \  { N } ) N  e.  ( <. V ,  E >. Neighbors 
v ) )
Distinct variable groups:    v, V    v, E    v, N

Proof of Theorem uvtxnbgravtx
StepHypRef Expression
1 uvtxnbgra 23538 . . 3  |-  ( ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
)  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } ) )
2 eleq2 2524 . . . . . . 7  |-  ( (
<. V ,  E >. Neighbors  N
)  =  ( V 
\  { N }
)  ->  ( v  e.  ( <. V ,  E >. Neighbors  N )  <->  v  e.  ( V  \  { N } ) ) )
32adantr 465 . . . . . 6  |-  ( ( ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } )  /\  ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
) )  ->  (
v  e.  ( <. V ,  E >. Neighbors  N
)  <->  v  e.  ( V  \  { N } ) ) )
43biimpar 485 . . . . 5  |-  ( ( ( ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } )  /\  ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
) )  /\  v  e.  ( V  \  { N } ) )  -> 
v  e.  ( <. V ,  E >. Neighbors  N
) )
5 nbgrasym 23485 . . . . . . 7  |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors 
v )  <->  v  e.  ( <. V ,  E >. Neighbors  N ) ) )
65ad2antrl 727 . . . . . 6  |-  ( ( ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } )  /\  ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
) )  ->  ( N  e.  ( <. V ,  E >. Neighbors  v )  <-> 
v  e.  ( <. V ,  E >. Neighbors  N
) ) )
76adantr 465 . . . . 5  |-  ( ( ( ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } )  /\  ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
) )  /\  v  e.  ( V  \  { N } ) )  -> 
( N  e.  (
<. V ,  E >. Neighbors  v
)  <->  v  e.  (
<. V ,  E >. Neighbors  N
) ) )
84, 7mpbird 232 . . . 4  |-  ( ( ( ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } )  /\  ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
) )  /\  v  e.  ( V  \  { N } ) )  ->  N  e.  ( <. V ,  E >. Neighbors  v ) )
98ex 434 . . 3  |-  ( ( ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } )  /\  ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
) )  ->  (
v  e.  ( V 
\  { N }
)  ->  N  e.  ( <. V ,  E >. Neighbors 
v ) ) )
101, 9mpancom 669 . 2  |-  ( ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
)  ->  ( v  e.  ( V  \  { N } )  ->  N  e.  ( <. V ,  E >. Neighbors 
v ) ) )
1110ralrimiv 2820 1  |-  ( ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
)  ->  A. v  e.  ( V  \  { N } ) N  e.  ( <. V ,  E >. Neighbors 
v ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    \ cdif 3425   {csn 3977   <.cop 3983   class class class wbr 4392  (class class class)co 6192   USGrph cusg 23401   Neighbors cnbgra 23466   UnivVertex cuvtx 23468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  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 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-cda 8440  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-hash 12207  df-usgra 23403  df-nbgra 23469  df-uvtx 23471
This theorem is referenced by: (None)
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