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Theorem efrn2lp 4565
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 4561 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B  _E  C  /\  C  _E  B
) )
2 epelg 4496 . . . 4  |-  ( C  e.  A  ->  ( B  _E  C  <->  B  e.  C ) )
3 epelg 4496 . . . 4  |-  ( B  e.  A  ->  ( C  _E  B  <->  C  e.  B ) )
42, 3bi2anan9r 846 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( ( B  _E  C  /\  C  _E  B
)  <->  ( B  e.  C  /\  C  e.  B ) ) )
54adantl 454 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B  _E  C  /\  C  _E  B
)  <->  ( B  e.  C  /\  C  e.  B ) ) )
61, 5mtbid 293 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 178    /\ wa 360    e. wcel 1726   class class class wbr 4213    _E cep 4493    Fr wfr 4539
This theorem is referenced by:  en2lp  7572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-eprel 4495  df-fr 4542
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