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Theorem 3v3e3cycl 24341
Description: If and only if there is a 3-cycle in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
Assertion
Ref Expression
3v3e3cycl  |-  ( V USGrph  E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3 )  <->  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
) ) )
Distinct variable groups:    E, a,
b, c, f, p    V, a, b, c, f, p

Proof of Theorem 3v3e3cycl
StepHypRef Expression
1 usgrafun 24025 . . 3  |-  ( V USGrph  E  ->  Fun  E )
2 19.41vv 1946 . . . . 5  |-  ( E. f E. p ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  <->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E
) )
3 simpr 461 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  Fun  E )
4 simpll 753 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  f
( V Cycles  E )
p )
5 simplr 754 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  ( # `
 f )  =  3 )
6 3v3e3cycl1 24320 . . . . . . 7  |-  ( ( Fun  E  /\  f
( V Cycles  E )
p  /\  ( # `  f
)  =  3 )  ->  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
) )
73, 4, 5, 6syl3anc 1228 . . . . . 6  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  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 ) )
87exlimivv 1699 . . . . 5  |-  ( E. f E. p ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  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 ) )
92, 8sylbir 213 . . . 4  |-  ( ( E. f E. p
( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  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 ) )
109expcom 435 . . 3  |-  ( Fun 
E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  ->  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 ) ) )
111, 10syl 16 . 2  |-  ( V USGrph  E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3 )  ->  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
) ) )
12 3v3e3cycl2 24340 . 2  |-  ( V USGrph  E  ->  ( 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 )  ->  E. f E. p
( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 ) ) )
1311, 12impbid 191 1  |-  ( V USGrph  E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3 )  <->  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
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815   {cpr 4029   class class class wbr 4447   ran crn 5000   Fun wfun 5580   ` cfv 5586  (class class class)co 6282   3c3 10582   #chash 12369   USGrph cusg 24006   Cycles ccycl 24183
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 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504  df-usgra 24009  df-wlk 24184  df-trail 24185  df-pth 24186  df-cycl 24189
This theorem is referenced by: (None)
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