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Theorem 3cyclfrgrarn 24786
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 24765 . . . 4  |-  ( V FriendGrph  E  ->  V USGrph  E )
2 usgrav 24111 . . . 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 12448 . . . . . . . . . . . . . 14  |-  ( ( V  e.  _V  /\  1  <  ( # `  V
)  /\  a  e.  V )  ->  E. x  e.  V  a  =/=  x )
543expa 1196 . . . . . . . . . . . . 13  |-  ( ( ( V  e.  _V  /\  1  <  ( # `  V ) )  /\  a  e.  V )  ->  E. x  e.  V  a  =/=  x )
6 3cyclfrgrarn1 24785 . . . . . . . . . . . . . . . . . . . 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 1197 . . . . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . . . 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 83 . . . . . . . . . . . . . 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 2949 . . . . . . . . . . . . 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 435 . . . . . . . . . . 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 87 . . . . . . . . . 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 47 . . . . . . . . 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 80 . . . . . . . 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 429 . . . . . . 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 2876 . . . . . 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 604 . . . . 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 78 . . . 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 465 . . 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 36 . 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 429 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113   {cpr 4029   class class class wbr 4447   ran crn 5000   ` cfv 5588   1c1 9494    < clt 9629   #chash 12374   USGrph cusg 24103   FriendGrph cfrgra 24761
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-hash 12375  df-usgra 24106  df-frgra 24762
This theorem is referenced by:  3cyclfrgrarn2  24787  3cyclfrgra  24788  vdn0frgrav2  24797  vdgn0frgrav2  24798
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