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Theorem efrn2lp 4867
Description: A set founded by epsilon contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994.)
Assertion
Ref Expression
efrn2lp  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B  e.  C  /\  C  e.  B
) )

Proof of Theorem efrn2lp
StepHypRef Expression
1 fr2nr 4863 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B  _E  C  /\  C  _E  B
) )
2 epelg 4798 . . . 4  |-  ( C  e.  A  ->  ( B  _E  C  <->  B  e.  C ) )
3 epelg 4798 . . . 4  |-  ( B  e.  A  ->  ( C  _E  B  <->  C  e.  B ) )
42, 3bi2anan9r 872 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( ( B  _E  C  /\  C  _E  B
)  <->  ( B  e.  C  /\  C  e.  B ) ) )
54adantl 466 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B  _E  C  /\  C  _E  B
)  <->  ( B  e.  C  /\  C  e.  B ) ) )
61, 5mtbid 300 1  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B  e.  C  /\  C  e.  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767   class class class wbr 4453    _E cep 4795    Fr wfr 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-eprel 4797  df-fr 4844
This theorem is referenced by:  en2lp  8042
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