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Theorem rusgrasn 30580
Description: If a k-regular undirected simple graph has only one vertex, then k must be 0. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
Assertion
Ref Expression
rusgrasn  |-  ( ( ( # `  V
)  =  1  /\ 
<. V ,  E >. RegUSGrph  K
)  ->  K  = 
0 )

Proof of Theorem rusgrasn
Dummy variables  k  n  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rusgraprop3 30578 . . 3  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. k  e.  V  ( # `
 { n  e.  V  |  { k ,  n }  e.  ran  E } )  =  K ) )
2 usgrav 23289 . . . . 5  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
3 hash1snb 12190 . . . . . . . 8  |-  ( V  e.  _V  ->  (
( # `  V )  =  1  <->  E. v  V  =  { v } ) )
4 raleq 2936 . . . . . . . . . . . . 13  |-  ( V  =  { v }  ->  ( A. k  e.  V  ( # `  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)  =  K  <->  A. k  e.  { v }  ( # `
 { n  e.  V  |  { k ,  n }  e.  ran  E } )  =  K ) )
5 vex 2994 . . . . . . . . . . . . . 14  |-  v  e. 
_V
6 preq1 3973 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  v  ->  { k ,  n }  =  { v ,  n } )
76eleq1d 2509 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  v  ->  ( { k ,  n }  e.  ran  E  <->  { v ,  n }  e.  ran  E ) )
87rabbidv 2983 . . . . . . . . . . . . . . . . 17  |-  ( k  =  v  ->  { n  e.  V  |  {
k ,  n }  e.  ran  E }  =  { n  e.  V  |  { v ,  n }  e.  ran  E }
)
98fveq2d 5714 . . . . . . . . . . . . . . . 16  |-  ( k  =  v  ->  ( # `
 { n  e.  V  |  { k ,  n }  e.  ran  E } )  =  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
) )
109eqeq1d 2451 . . . . . . . . . . . . . . 15  |-  ( k  =  v  ->  (
( # `  { n  e.  V  |  {
k ,  n }  e.  ran  E } )  =  K  <->  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K ) )
1110ralsng 3931 . . . . . . . . . . . . . 14  |-  ( v  e.  _V  ->  ( A. k  e.  { v }  ( # `  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)  =  K  <->  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K ) )
125, 11mp1i 12 . . . . . . . . . . . . 13  |-  ( V  =  { v }  ->  ( A. k  e.  { v }  ( # `
 { n  e.  V  |  { k ,  n }  e.  ran  E } )  =  K  <->  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K ) )
13 rabeq 2985 . . . . . . . . . . . . . . 15  |-  ( V  =  { v }  ->  { n  e.  V  |  { v ,  n }  e.  ran  E }  =  {
n  e.  { v }  |  { v ,  n }  e.  ran  E } )
1413fveq2d 5714 . . . . . . . . . . . . . 14  |-  ( V  =  { v }  ->  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  ( # `  { n  e.  {
v }  |  {
v ,  n }  e.  ran  E } ) )
1514eqeq1d 2451 . . . . . . . . . . . . 13  |-  ( V  =  { v }  ->  ( ( # `  { n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K  <->  ( # `  {
n  e.  { v }  |  { v ,  n }  e.  ran  E } )  =  K ) )
164, 12, 153bitrd 279 . . . . . . . . . . . 12  |-  ( V  =  { v }  ->  ( A. k  e.  V  ( # `  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)  =  K  <->  ( # `  {
n  e.  { v }  |  { v ,  n }  e.  ran  E } )  =  K ) )
17 hashrabsn01 30255 . . . . . . . . . . . . . 14  |-  ( (
# `  { n  e.  { v }  |  { v ,  n }  e.  ran  E }
)  =  K  -> 
( K  =  0  \/  K  =  1 ) )
18 ax-1 6 . . . . . . . . . . . . . . . . . . 19  |-  ( K  =  0  ->  (
( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) )
1918a1d 25 . . . . . . . . . . . . . . . . . 18  |-  ( K  =  0  ->  ( V  e.  _V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) ) )
2019a1d 25 . . . . . . . . . . . . . . . . 17  |-  ( K  =  0  ->  ( K  e.  NN0  ->  ( V  e.  _V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) ) ) )
2120a1d 25 . . . . . . . . . . . . . . . 16  |-  ( K  =  0  ->  ( V  =  { v }  ->  ( K  e. 
NN0  ->  ( V  e. 
_V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  = 
0 ) ) ) ) )
2221a1d 25 . . . . . . . . . . . . . . 15  |-  ( K  =  0  ->  (
( # `  { n  e.  { v }  |  { v ,  n }  e.  ran  E }
)  =  K  -> 
( V  =  {
v }  ->  ( K  e.  NN0  ->  ( V  e.  _V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) ) ) ) ) )
23 eqeq2 2452 . . . . . . . . . . . . . . . 16  |-  ( K  =  1  ->  (
( # `  { n  e.  { v }  |  { v ,  n }  e.  ran  E }
)  =  K  <->  ( # `  {
n  e.  { v }  |  { v ,  n }  e.  ran  E } )  =  1 ) )
24 hashrabsn1 30256 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  { n  e.  { v }  |  { v ,  n }  e.  ran  E }
)  =  1  ->  [. v  /  n ]. { v ,  n }  e.  ran  E )
25 sbcel1g 3700 . . . . . . . . . . . . . . . . . . 19  |-  ( v  e.  _V  ->  ( [. v  /  n ]. { v ,  n }  e.  ran  E  <->  [_ v  /  n ]_ { v ,  n }  e.  ran  E ) )
265, 25ax-mp 5 . . . . . . . . . . . . . . . . . 18  |-  ( [. v  /  n ]. {
v ,  n }  e.  ran  E  <->  [_ v  /  n ]_ { v ,  n }  e.  ran  E )
27 csbprg 30140 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( v  e.  _V  ->  [_ v  /  n ]_ { v ,  n }  =  { [_ v  /  n ]_ v ,  [_ v  /  n ]_ n }
)
28 csbconstg 3320 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( v  e.  _V  ->  [_ v  /  n ]_ v  =  v )
29 csbvarg 3719 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( v  e.  _V  ->  [_ v  /  n ]_ n  =  v )
3028, 29preq12d 3981 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( v  e.  _V  ->  { [_ v  /  n ]_ v ,  [_ v  /  n ]_ n }  =  {
v ,  v } )
3127, 30eqtrd 2475 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  e.  _V  ->  [_ v  /  n ]_ { v ,  n }  =  { v ,  v } )
325, 31ax-mp 5 . . . . . . . . . . . . . . . . . . . 20  |-  [_ v  /  n ]_ { v ,  n }  =  { v ,  v }
3332eleq1i 2506 . . . . . . . . . . . . . . . . . . 19  |-  ( [_ v  /  n ]_ {
v ,  n }  e.  ran  E  <->  { v ,  v }  e.  ran  E )
34 equid 1729 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  v  =  v
35 usgraedgrn 23319 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( V USGrph  E  /\  { v ,  v }  e.  ran  E )  ->  v  =/=  v )
36 eqneqall 2724 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( v  =  v  ->  (
v  =/=  v  ->  K  =  0 ) )
3734, 35, 36mpsyl 63 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( V USGrph  E  /\  { v ,  v }  e.  ran  E )  ->  K  =  0 )
3837ex 434 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( V USGrph  E  ->  ( { v ,  v }  e.  ran  E  ->  K  = 
0 ) )
3938adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( E  e.  _V  /\  V USGrph  E )  ->  ( { v ,  v }  e.  ran  E  ->  K  =  0 ) )
4039com12 31 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( { v ,  v }  e.  ran  E  -> 
( ( E  e. 
_V  /\  V USGrph  E )  ->  K  =  0 ) )
4140a1d 25 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { v ,  v }  e.  ran  E  -> 
( V  e.  _V  ->  ( ( E  e. 
_V  /\  V USGrph  E )  ->  K  =  0 ) ) )
4241a1d 25 . . . . . . . . . . . . . . . . . . . 20  |-  ( { v ,  v }  e.  ran  E  -> 
( K  e.  NN0  ->  ( V  e.  _V  ->  ( ( E  e. 
_V  /\  V USGrph  E )  ->  K  =  0 ) ) ) )
4342a1d 25 . . . . . . . . . . . . . . . . . . 19  |-  ( { v ,  v }  e.  ran  E  -> 
( V  =  {
v }  ->  ( K  e.  NN0  ->  ( V  e.  _V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) ) ) ) )
4433, 43sylbi 195 . . . . . . . . . . . . . . . . . 18  |-  ( [_ v  /  n ]_ {
v ,  n }  e.  ran  E  ->  ( V  =  { v }  ->  ( K  e. 
NN0  ->  ( V  e. 
_V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  = 
0 ) ) ) ) )
4526, 44sylbi 195 . . . . . . . . . . . . . . . . 17  |-  ( [. v  /  n ]. {
v ,  n }  e.  ran  E  ->  ( V  =  { v }  ->  ( K  e. 
NN0  ->  ( V  e. 
_V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  = 
0 ) ) ) ) )
4624, 45syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
# `  { n  e.  { v }  |  { v ,  n }  e.  ran  E }
)  =  1  -> 
( V  =  {
v }  ->  ( K  e.  NN0  ->  ( V  e.  _V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) ) ) ) )
4723, 46syl6bi 228 . . . . . . . . . . . . . . 15  |-  ( K  =  1  ->  (
( # `  { n  e.  { v }  |  { v ,  n }  e.  ran  E }
)  =  K  -> 
( V  =  {
v }  ->  ( K  e.  NN0  ->  ( V  e.  _V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) ) ) ) ) )
4822, 47jaoi 379 . . . . . . . . . . . . . 14  |-  ( ( K  =  0  \/  K  =  1 )  ->  ( ( # `  { n  e.  {
v }  |  {
v ,  n }  e.  ran  E } )  =  K  ->  ( V  =  { v }  ->  ( K  e. 
NN0  ->  ( V  e. 
_V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  = 
0 ) ) ) ) ) )
4917, 48mpcom 36 . . . . . . . . . . . . 13  |-  ( (
# `  { n  e.  { v }  |  { v ,  n }  e.  ran  E }
)  =  K  -> 
( V  =  {
v }  ->  ( K  e.  NN0  ->  ( V  e.  _V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) ) ) ) )
5049com12 31 . . . . . . . . . . . 12  |-  ( V  =  { v }  ->  ( ( # `  { n  e.  {
v }  |  {
v ,  n }  e.  ran  E } )  =  K  ->  ( K  e.  NN0  ->  ( V  e.  _V  ->  ( ( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) ) ) ) )
5116, 50sylbid 215 . . . . . . . . . . 11  |-  ( V  =  { v }  ->  ( A. k  e.  V  ( # `  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)  =  K  -> 
( K  e.  NN0  ->  ( V  e.  _V  ->  ( ( E  e. 
_V  /\  V USGrph  E )  ->  K  =  0 ) ) ) ) )
5251com24 87 . . . . . . . . . 10  |-  ( V  =  { v }  ->  ( V  e. 
_V  ->  ( K  e. 
NN0  ->  ( A. k  e.  V  ( # `  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)  =  K  -> 
( ( E  e. 
_V  /\  V USGrph  E )  ->  K  =  0 ) ) ) ) )
5352exlimiv 1688 . . . . . . . . 9  |-  ( E. v  V  =  {
v }  ->  ( V  e.  _V  ->  ( K  e.  NN0  ->  ( A. k  e.  V  ( # `  { n  e.  V  |  {
k ,  n }  e.  ran  E } )  =  K  ->  (
( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) ) ) ) )
5453com12 31 . . . . . . . 8  |-  ( V  e.  _V  ->  ( E. v  V  =  { v }  ->  ( K  e.  NN0  ->  ( A. k  e.  V  ( # `  { n  e.  V  |  {
k ,  n }  e.  ran  E } )  =  K  ->  (
( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) ) ) ) )
553, 54sylbid 215 . . . . . . 7  |-  ( V  e.  _V  ->  (
( # `  V )  =  1  ->  ( K  e.  NN0  ->  ( A. k  e.  V  ( # `  { n  e.  V  |  {
k ,  n }  e.  ran  E } )  =  K  ->  (
( E  e.  _V  /\  V USGrph  E )  ->  K  =  0 ) ) ) ) )
5655com25 91 . . . . . 6  |-  ( V  e.  _V  ->  (
( E  e.  _V  /\  V USGrph  E )  -> 
( K  e.  NN0  ->  ( A. k  e.  V  ( # `  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)  =  K  -> 
( ( # `  V
)  =  1  ->  K  =  0 ) ) ) ) )
5756expdimp 437 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V USGrph  E  -> 
( K  e.  NN0  ->  ( A. k  e.  V  ( # `  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)  =  K  -> 
( ( # `  V
)  =  1  ->  K  =  0 ) ) ) ) )
582, 57mpcom 36 . . . 4  |-  ( V USGrph  E  ->  ( K  e. 
NN0  ->  ( A. k  e.  V  ( # `  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)  =  K  -> 
( ( # `  V
)  =  1  ->  K  =  0 ) ) ) )
59583imp 1181 . . 3  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. k  e.  V  ( # `  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)  =  K )  ->  ( ( # `  V )  =  1  ->  K  =  0 ) )
601, 59syl 16 . 2  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( ( # `  V
)  =  1  ->  K  =  0 ) )
6160impcom 430 1  |-  ( ( ( # `  V
)  =  1  /\ 
<. V ,  E >. RegUSGrph  K
)  ->  K  = 
0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2620   A.wral 2734   {crab 2738   _Vcvv 2991   [.wsbc 3205   [_csb 3307   {csn 3896   {cpr 3898   <.cop 3902   class class class wbr 4311   ran crn 4860   ` cfv 5437   0cc0 9301   1c1 9302   NN0cn0 10598   #chash 12122   USGrph cusg 23283   RegUSGrph crusgra 30563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-2o 6940  df-oadd 6943  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-card 8128  df-cda 8356  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-2 10399  df-n0 10599  df-z 10666  df-uz 10881  df-xadd 11109  df-fz 11457  df-hash 12123  df-usgra 23285  df-nbgra 23351  df-vdgr 23583  df-rgra 30564  df-rusgra 30565
This theorem is referenced by:  frgrareg  30733
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