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Theorem epne3 4764
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 4763 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B  _E  C  /\  C  _E  D  /\  D  _E  B
) )
2 epelg 4498 . . . . 5  |-  ( C  e.  A  ->  ( B  _E  C  <->  B  e.  C ) )
323ad2ant2 980 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( B  _E  C  <->  B  e.  C ) )
4 epelg 4498 . . . . 5  |-  ( D  e.  A  ->  ( C  _E  D  <->  C  e.  D ) )
543ad2ant3 981 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( C  _E  D  <->  C  e.  D ) )
6 epelg 4498 . . . . 5  |-  ( B  e.  A  ->  ( D  _E  B  <->  D  e.  B ) )
763ad2ant1 979 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( D  _E  B  <->  D  e.  B ) )
83, 5, 73anbi123d 1255 . . 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 454 . 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 293 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 178    /\ wa 360    /\ w3a 937    e. wcel 1726   class class class wbr 4215    _E cep 4495    Fr wfr 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
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-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-eprel 4497  df-fr 4544
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