MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  en3lp Structured version   Unicode version

Theorem en3lp 7818
Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 31415 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 3638 . . . . 5  |-  -.  C  e.  (/)
2 eleq2 2502 . . . . 5  |-  ( { A ,  B ,  C }  =  (/)  ->  ( C  e.  { A ,  B ,  C }  <->  C  e.  (/) ) )
31, 2mtbiri 303 . . . 4  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  C  e.  { A ,  B ,  C }
)
4 tpid3g 3987 . . . 4  |-  ( C  e.  A  ->  C  e.  { A ,  B ,  C } )
53, 4nsyl 121 . . 3  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  C  e.  A )
6 simp3 985 . . 3  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  A )
75, 6nsyl 121 . 2  |-  ( { A ,  B ,  C }  =  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A
) )
8 tpex 6378 . . . 4  |-  { A ,  B ,  C }  e.  _V
98zfreg 7806 . . 3  |-  ( { A ,  B ,  C }  =/=  (/)  ->  E. x  e.  { A ,  B ,  C }  ( x  i^i  { A ,  B ,  C }
)  =  (/) )
10 en3lplem2 7817 . . . . . 6  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  i^i  { A ,  B ,  C }
)  =/=  (/) ) )
1110com12 31 . . . . 5  |-  ( x  e.  { A ,  B ,  C }  ->  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) ) )
1211necon2bd 2658 . . . 4  |-  ( x  e.  { A ,  B ,  C }  ->  ( ( x  i^i 
{ A ,  B ,  C } )  =  (/)  ->  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) ) )
1312rexlimiv 2833 . . 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 2685 1  |-  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714    i^i cin 3324   (/)c0 3634   {ctp 3878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371  ax-reg 7803
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-nul 3635  df-sn 3875  df-pr 3877  df-tp 3879  df-uni 4089
This theorem is referenced by:  tratrb  31075  tratrbVD  31431  bj-inftyexpidisj  32262
  Copyright terms: Public domain W3C validator