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Theorem en3lp 7674
Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 29019 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 3634 . . . . 5  |-  -.  C  e.  (/)
2 eleq2 2499 . . . . 5  |-  ( { A ,  B ,  C }  =  (/)  ->  ( C  e.  { A ,  B ,  C }  <->  C  e.  (/) ) )
31, 2mtbiri 296 . . . 4  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  C  e.  { A ,  B ,  C }
)
4 tpid3g 3921 . . . 4  |-  ( C  e.  A  ->  C  e.  { A ,  B ,  C } )
53, 4nsyl 116 . . 3  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  C  e.  A )
6 simp3 960 . . 3  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  A )
75, 6nsyl 116 . 2  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A
) )
8 tpex 4710 . . . 4  |-  { A ,  B ,  C }  e.  _V
98zfreg 7565 . . 3  |-  ( { A ,  B ,  C }  =/=  (/)  ->  E. x  e.  { A ,  B ,  C }  ( x  i^i  { A ,  B ,  C }
)  =  (/) )
10 en3lplem2 7673 . . . . . 6  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  i^i  { A ,  B ,  C }
)  =/=  (/) ) )
1110com12 30 . . . . 5  |-  ( x  e.  { A ,  B ,  C }  ->  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) ) )
1211necon2bd 2655 . . . 4  |-  ( x  e.  { A ,  B ,  C }  ->  ( ( x  i^i 
{ A ,  B ,  C } )  =  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ) )
1312rexlimiv 2826 . . 3  |-  ( E. x  e.  { A ,  B ,  C } 
( x  i^i  { A ,  B ,  C } )  =  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) )
149, 13syl 16 . 2  |-  ( { A ,  B ,  C }  =/=  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A
) )
157, 14pm2.61ine 2682 1  |-  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    i^i cin 3321   (/)c0 3630   {ctp 3818
This theorem is referenced by:  tratrb  28682  tratrbVD  29035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703  ax-reg 7562
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-nul 3631  df-sn 3822  df-pr 3823  df-tp 3824  df-uni 4018
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