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Theorem 3cycl3dv 21612
Description: In a simple graph, the vertices of a 3-cycle are mutually different. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
Assertion
Ref Expression
3cycl3dv  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( A  =/= 
B  /\  B  =/=  C  /\  C  =/=  A
) )

Proof of Theorem 3cycl3dv
StepHypRef Expression
1 usgraedgrn 21384 . . . 4  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  A  =/=  B )
21ex 424 . . 3  |-  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  ->  A  =/=  B
) )
3 usgraedgrn 21384 . . . 4  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  B  =/=  C )
43ex 424 . . 3  |-  ( V USGrph  E  ->  ( { B ,  C }  e.  ran  E  ->  B  =/=  C
) )
5 usgraedgrn 21384 . . . 4  |-  ( ( V USGrph  E  /\  { C ,  A }  e.  ran  E )  ->  C  =/=  A )
65ex 424 . . 3  |-  ( V USGrph  E  ->  ( { C ,  A }  e.  ran  E  ->  C  =/=  A
) )
72, 4, 63anim123d 1261 . 2  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  -> 
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) ) )
87imp 419 1  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( A  =/= 
B  /\  B  =/=  C  /\  C  =/=  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2593   {cpr 3802   class class class wbr 4199   ran crn 4865   USGrph cusg 21348
This theorem is referenced by:  constr3trllem2  21621  constr3pth  21630
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 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051
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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-oadd 6714  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-card 7810  df-cda 8032  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-nn 9985  df-2 10042  df-n0 10206  df-z 10267  df-uz 10473  df-fz 11028  df-hash 11602  df-usgra 21350
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