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Theorem efrn2lp 4813
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 4809 . 2  |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B  _E  C  /\  C  _E  B
) )
2 epelg 4744 . . . 4  |-  ( C  e.  A  ->  ( B  _E  C  <->  B  e.  C ) )
3 epelg 4744 . . . 4  |-  ( B  e.  A  ->  ( C  _E  B  <->  C  e.  B ) )
42, 3bi2anan9r 869 . . 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 1758   class class class wbr 4403    _E cep 4741    Fr wfr 4787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-eprel 4743  df-fr 4790
This theorem is referenced by:  en2lp  7933
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