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Theorem epne3 6601
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 6600 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B  _E  C  /\  C  _E  D  /\  D  _E  B
) )
2 epelg 4782 . . . . 5  |-  ( C  e.  A  ->  ( B  _E  C  <->  B  e.  C ) )
323ad2ant2 1019 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( B  _E  C  <->  B  e.  C ) )
4 epelg 4782 . . . . 5  |-  ( D  e.  A  ->  ( C  _E  D  <->  C  e.  D ) )
543ad2ant3 1020 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( C  _E  D  <->  C  e.  D ) )
6 epelg 4782 . . . . 5  |-  ( B  e.  A  ->  ( D  _E  B  <->  D  e.  B ) )
763ad2ant1 1018 . . . 4  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( D  _E  B  <->  D  e.  B ) )
83, 5, 73anbi123d 1300 . . 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 466 . 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 300 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    e. wcel 1804   class class class wbr 4437    _E cep 4779    Fr wfr 4825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-eprel 4781  df-fr 4828
This theorem is referenced by: (None)
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