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Theorem en3lp 8121
Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 36881 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 3771 . . . . 5  |-  -.  C  e.  (/)
2 eleq2 2502 . . . . 5  |-  ( { A ,  B ,  C }  =  (/)  ->  ( C  e.  { A ,  B ,  C }  <->  C  e.  (/) ) )
31, 2mtbiri 304 . . . 4  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  C  e.  { A ,  B ,  C }
)
4 tpid3g 4118 . . . 4  |-  ( C  e.  A  ->  C  e.  { A ,  B ,  C } )
53, 4nsyl 124 . . 3  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  C  e.  A )
65intn3an3d 1376 . 2  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A
) )
7 tpex 6604 . . . 4  |-  { A ,  B ,  C }  e.  _V
87zfreg 8110 . . 3  |-  ( { A ,  B ,  C }  =/=  (/)  ->  E. x  e.  { A ,  B ,  C }  ( x  i^i  { A ,  B ,  C }
)  =  (/) )
9 en3lplem2 8120 . . . . . 6  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  i^i  { A ,  B ,  C }
)  =/=  (/) ) )
109com12 32 . . . . 5  |-  ( x  e.  { A ,  B ,  C }  ->  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) ) )
1110necon2bd 2646 . . . 4  |-  ( x  e.  { A ,  B ,  C }  ->  ( ( x  i^i 
{ A ,  B ,  C } )  =  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ) )
1211rexlimiv 2918 . . 3  |-  ( E. x  e.  { A ,  B ,  C } 
( x  i^i  { A ,  B ,  C } )  =  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) )
138, 12syl 17 . 2  |-  ( { A ,  B ,  C }  =/=  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A
) )
146, 13pm2.61ine 2744 1  |-  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783    i^i cin 3441   (/)c0 3767   {ctp 4006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597  ax-reg 8107
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-nul 3768  df-sn 4003  df-pr 4005  df-tp 4007  df-uni 4223
This theorem is referenced by:  bj-inftyexpidisj  31397  tratrb  36534  tratrbVD  36898
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