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Theorem en3lp 7418
Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 28621 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
en3lp  |-  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A )

Proof of Theorem en3lp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 noel 3459 . . . . 5  |-  -.  C  e.  (/)
2 eleq2 2344 . . . . 5  |-  ( { A ,  B ,  C }  =  (/)  ->  ( C  e.  { A ,  B ,  C }  <->  C  e.  (/) ) )
31, 2mtbiri 294 . . . 4  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  C  e.  { A ,  B ,  C }
)
4 tpid3g 3741 . . . 4  |-  ( C  e.  A  ->  C  e.  { A ,  B ,  C } )
53, 4nsyl 113 . . 3  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  C  e.  A )
6 simp3 957 . . 3  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  A )
75, 6nsyl 113 . 2  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A
) )
8 tpex 4519 . . . 4  |-  { A ,  B ,  C }  e.  _V
98zfreg 7309 . . 3  |-  ( { A ,  B ,  C }  =/=  (/)  ->  E. x  e.  { A ,  B ,  C }  ( x  i^i  { A ,  B ,  C }
)  =  (/) )
10 en3lplem2 7417 . . . . . 6  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  i^i  { A ,  B ,  C }
)  =/=  (/) ) )
1110com12 27 . . . . 5  |-  ( x  e.  { A ,  B ,  C }  ->  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) ) )
1211necon2bd 2495 . . . 4  |-  ( x  e.  { A ,  B ,  C }  ->  ( ( x  i^i 
{ A ,  B ,  C } )  =  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ) )
1312rexlimiv 2661 . . 3  |-  ( E. x  e.  { A ,  B ,  C } 
( x  i^i  { A ,  B ,  C } )  =  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) )
149, 13syl 15 . 2  |-  ( { A ,  B ,  C }  =/=  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A
) )
157, 14pm2.61ine 2522 1  |-  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    i^i cin 3151   (/)c0 3455   {ctp 3642
This theorem is referenced by:  tratrb  28299  tratrbVD  28637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-nul 3456  df-sn 3646  df-pr 3647  df-tp 3648  df-uni 3828
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