MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  epne3 Unicode version

Theorem epne3 4728
Description: A set well-founded by epsilon contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
epne3  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B  e.  C  /\  C  e.  D  /\  D  e.  B
) )

Proof of Theorem epne3
StepHypRef Expression
1 fr3nr 4727 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B  _E  C  /\  C  _E  D  /\  D  _E  B
) )
2 epelg 4463 . . . . 5  |-  ( C  e.  A  ->  ( B  _E  C  <->  B  e.  C ) )
323ad2ant2 979 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( B  _E  C  <->  B  e.  C ) )
4 epelg 4463 . . . . 5  |-  ( D  e.  A  ->  ( C  _E  D  <->  C  e.  D ) )
543ad2ant3 980 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( C  _E  D  <->  C  e.  D ) )
6 epelg 4463 . . . . 5  |-  ( B  e.  A  ->  ( D  _E  B  <->  D  e.  B ) )
763ad2ant1 978 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( D  _E  B  <->  D  e.  B ) )
83, 5, 73anbi123d 1254 . . 3  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( ( B  _E  C  /\  C  _E  D  /\  D  _E  B
)  <->  ( B  e.  C  /\  C  e.  D  /\  D  e.  B ) ) )
98adantl 453 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B  _E  C  /\  C  _E  D  /\  D  _E  B
)  <->  ( B  e.  C  /\  C  e.  D  /\  D  e.  B ) ) )
101, 9mtbid 292 1  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B  e.  C  /\  C  e.  D  /\  D  e.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1721   class class class wbr 4180    _E cep 4460    Fr wfr 4506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-eprel 4462  df-fr 4509
  Copyright terms: Public domain W3C validator