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Theorem rusgrasn 24768
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 24766 . . 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 24161 . . . . 5  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
3 hash1snb 12459 . . . . . . . 8  |-  ( V  e.  _V  ->  (
( # `  V )  =  1  <->  E. v  V  =  { v } ) )
4 raleq 3063 . . . . . . . . . . . . 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 3121 . . . . . . . . . . . . . 14  |-  v  e. 
_V
6 preq1 4112 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  v  ->  { k ,  n }  =  { v ,  n } )
76eleq1d 2536 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  v  ->  ( { k ,  n }  e.  ran  E  <->  { v ,  n }  e.  ran  E ) )
87rabbidv 3110 . . . . . . . . . . . . . . . . 17  |-  ( k  =  v  ->  { n  e.  V  |  {
k ,  n }  e.  ran  E }  =  { n  e.  V  |  { v ,  n }  e.  ran  E }
)
98fveq2d 5876 . . . . . . . . . . . . . . . 16  |-  ( k  =  v  ->  ( # `
 { n  e.  V  |  { k ,  n }  e.  ran  E } )  =  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
) )
109eqeq1d 2469 . . . . . . . . . . . . . . 15  |-  ( k  =  v  ->  (
( # `  { n  e.  V  |  {
k ,  n }  e.  ran  E } )  =  K  <->  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K ) )
1110ralsng 4068 . . . . . . . . . . . . . 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 3112 . . . . . . . . . . . . . . 15  |-  ( V  =  { v }  ->  { n  e.  V  |  { v ,  n }  e.  ran  E }  =  {
n  e.  { v }  |  { v ,  n }  e.  ran  E } )
1413fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( V  =  { v }  ->  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  ( # `  { n  e.  {
v }  |  {
v ,  n }  e.  ran  E } ) )
1514eqeq1d 2469 . . . . . . . . . . . . 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 12421 . . . . . . . . . . . . . 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 2482 . . . . . . . . . . . . . . . 16  |-  ( K  =  1  ->  (
( # `  { n  e.  { v }  |  { v ,  n }  e.  ran  E }
)  =  K  <->  ( # `  {
n  e.  { v }  |  { v ,  n }  e.  ran  E } )  =  1 ) )
24 hashrabsn1 12422 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  { n  e.  { v }  |  { v ,  n }  e.  ran  E }
)  =  1  ->  [. v  /  n ]. { v ,  n }  e.  ran  E )
25 sbcel1g 3834 . . . . . . . . . . . . . . . . . . 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 4093 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( v  e.  _V  ->  [_ v  /  n ]_ { v ,  n }  =  { [_ v  /  n ]_ v ,  [_ v  /  n ]_ n }
)
28 csbconstg 3453 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( v  e.  _V  ->  [_ v  /  n ]_ v  =  v )
29 csbvarg 3853 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( v  e.  _V  ->  [_ v  /  n ]_ n  =  v )
3028, 29preq12d 4120 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( v  e.  _V  ->  { [_ v  /  n ]_ v ,  [_ v  /  n ]_ n }  =  {
v ,  v } )
3127, 30eqtrd 2508 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  e.  _V  ->  [_ v  /  n ]_ { v ,  n }  =  { v ,  v } )
325, 31ax-mp 5 . . . . . . . . . . . . . . . . . . . 20  |-  [_ v  /  n ]_ { v ,  n }  =  { v ,  v }
3332eleq1i 2544 . . . . . . . . . . . . . . . . . . 19  |-  ( [_ v  /  n ]_ {
v ,  n }  e.  ran  E  <->  { v ,  v }  e.  ran  E )
34 equid 1740 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  v  =  v
35 usgraedgrn 24204 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( V USGrph  E  /\  { v ,  v }  e.  ran  E )  ->  v  =/=  v )
36 eqneqall 2674 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1698 . . . . . . . . 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 1190 . . 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 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2817   {crab 2821   _Vcvv 3118   [.wsbc 3336   [_csb 3440   {csn 4033   {cpr 4035   <.cop 4039   class class class wbr 4453   ran crn 5006   ` cfv 5594   0cc0 9504   1c1 9505   NN0cn0 10807   #chash 12385   USGrph cusg 24153   RegUSGrph crusgra 24746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-xadd 11331  df-fz 11685  df-hash 12386  df-usgra 24156  df-nbgra 24243  df-vdgr 24717  df-rgra 24747  df-rusgra 24748
This theorem is referenced by:  frgrareg  24941
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