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Theorem efrn2lp 4375
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 4371 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B  _E  C  /\  C  _E  B
) )
2 epelg 4306 . . . 4  |-  ( C  e.  A  ->  ( B  _E  C  <->  B  e.  C ) )
3 epelg 4306 . . . 4  |-  ( B  e.  A  ->  ( C  _E  B  <->  C  e.  B ) )
42, 3bi2anan9r 844 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( ( B  _E  C  /\  C  _E  B
)  <->  ( B  e.  C  /\  C  e.  B ) ) )
54adantl 452 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B  _E  C  /\  C  _E  B
)  <->  ( B  e.  C  /\  C  e.  B ) ) )
61, 5mtbid 291 1  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B  e.  C  /\  C  e.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   class class class wbr 4023    _E cep 4303    Fr wfr 4349
This theorem is referenced by:  en2lp  7317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-fr 4352
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