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Theorem 3cyclfrgrarn 28304
Description: Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.)
Assertion
Ref Expression
3cyclfrgrarn  |-  ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
Distinct variable groups:    E, a,
b, c    V, a,
b, c

Proof of Theorem 3cyclfrgrarn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 frisusgra 28283 . . . 4  |-  ( V FriendGrph  E  ->  V USGrph  E )
2 usgrav 21361 . . . 4  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
31, 2syl 16 . . 3  |-  ( V FriendGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
4 hashgt12el2 11673 . . . . . . . . . . . . . 14  |-  ( ( V  e.  _V  /\  1  <  ( # `  V
)  /\  a  e.  V )  ->  E. x  e.  V  a  =/=  x )
543expa 1153 . . . . . . . . . . . . 13  |-  ( ( ( V  e.  _V  /\  1  <  ( # `  V ) )  /\  a  e.  V )  ->  E. x  e.  V  a  =/=  x )
6 3cyclfrgrarn1 28303 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V FriendGrph  E  /\  (
a  e.  V  /\  x  e.  V )  /\  a  =/=  x
)  ->  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
763expb 1154 . . . . . . . . . . . . . . . . . . 19  |-  ( ( V FriendGrph  E  /\  (
( a  e.  V  /\  x  e.  V
)  /\  a  =/=  x ) )  ->  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) )
87expcom 425 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  V  /\  x  e.  V
)  /\  a  =/=  x )  ->  ( V FriendGrph  E  ->  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) ) )
98ex 424 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  V  /\  x  e.  V )  ->  ( a  =/=  x  ->  ( V FriendGrph  E  ->  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) ) )
109expcom 425 . . . . . . . . . . . . . . . 16  |-  ( x  e.  V  ->  (
a  e.  V  -> 
( a  =/=  x  ->  ( V FriendGrph  E  ->  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) ) ) )
1110com23 74 . . . . . . . . . . . . . . 15  |-  ( x  e.  V  ->  (
a  =/=  x  -> 
( a  e.  V  ->  ( V FriendGrph  E  ->  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) ) ) )
1211com34 79 . . . . . . . . . . . . . 14  |-  ( x  e.  V  ->  (
a  =/=  x  -> 
( V FriendGrph  E  ->  (
a  e.  V  ->  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) ) ) )
1312rexlimiv 2816 . . . . . . . . . . . . 13  |-  ( E. x  e.  V  a  =/=  x  ->  ( V FriendGrph  E  ->  ( a  e.  V  ->  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) ) ) )
145, 13syl 16 . . . . . . . . . . . 12  |-  ( ( ( V  e.  _V  /\  1  <  ( # `  V ) )  /\  a  e.  V )  ->  ( V FriendGrph  E  ->  ( a  e.  V  ->  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) ) )
1514expcom 425 . . . . . . . . . . 11  |-  ( a  e.  V  ->  (
( V  e.  _V  /\  1  <  ( # `  V ) )  -> 
( V FriendGrph  E  ->  (
a  e.  V  ->  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) ) ) )
1615com24 83 . . . . . . . . . 10  |-  ( a  e.  V  ->  (
a  e.  V  -> 
( V FriendGrph  E  ->  (
( V  e.  _V  /\  1  <  ( # `  V ) )  ->  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) ) ) )
1716pm2.43i 45 . . . . . . . . 9  |-  ( a  e.  V  ->  ( V FriendGrph  E  ->  ( ( V  e.  _V  /\  1  <  ( # `  V
) )  ->  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) ) ) )
1817com13 76 . . . . . . . 8  |-  ( ( V  e.  _V  /\  1  <  ( # `  V
) )  ->  ( V FriendGrph  E  ->  ( a  e.  V  ->  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) ) ) )
1918imp 419 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  1  <  ( # `  V ) )  /\  V FriendGrph  E )  ->  (
a  e.  V  ->  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) )
2019ralrimiv 2780 . . . . . 6  |-  ( ( ( V  e.  _V  /\  1  <  ( # `  V ) )  /\  V FriendGrph  E )  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
2120exp31 588 . . . . 5  |-  ( V  e.  _V  ->  (
1  <  ( # `  V
)  ->  ( V FriendGrph  E  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) ) )
2221com23 74 . . . 4  |-  ( V  e.  _V  ->  ( V FriendGrph  E  ->  ( 1  <  ( # `  V
)  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) ) ) )
2322adantr 452 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V FriendGrph  E  ->  ( 1  <  ( # `  V )  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) ) ) )
243, 23mpcom 34 . 2  |-  ( V FriendGrph  E  ->  ( 1  < 
( # `  V )  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) )
2524imp 419 1  |-  ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948   {cpr 3807   class class class wbr 4204   ran crn 4871   ` cfv 5446   1c1 8981    < clt 9110   #chash 11608   USGrph cusg 21355   FriendGrph cfrgra 28279
This theorem is referenced by:  3cyclfrgrarn2  28305  3cyclfrgra  28306  vdn0frgrav2  28315  vdgn0frgrav2  28316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7816  df-cda 8038  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-hash 11609  df-usgra 21357  df-frgra 28280
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