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Theorem epne3 4654
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 4653 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B  _E  C  /\  C  _E  D  /\  D  _E  B
) )
2 epelg 4388 . . . . 5  |-  ( C  e.  A  ->  ( B  _E  C  <->  B  e.  C ) )
323ad2ant2 977 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( B  _E  C  <->  B  e.  C ) )
4 epelg 4388 . . . . 5  |-  ( D  e.  A  ->  ( C  _E  D  <->  C  e.  D ) )
543ad2ant3 978 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( C  _E  D  <->  C  e.  D ) )
6 epelg 4388 . . . . 5  |-  ( B  e.  A  ->  ( D  _E  B  <->  D  e.  B ) )
763ad2ant1 976 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( D  _E  B  <->  D  e.  B ) )
83, 5, 73anbi123d 1252 . . 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 452 . 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 291 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 176    /\ wa 358    /\ w3a 934    e. wcel 1710   class class class wbr 4104    _E cep 4385    Fr wfr 4431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-eprel 4387  df-fr 4434
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