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Theorem efrn2lp 5020
 Description: A set founded by epsilon contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994.)
Assertion
Ref Expression
efrn2lp (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝐶𝐶𝐵))

Proof of Theorem efrn2lp
StepHypRef Expression
1 fr2nr 5016 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵 E 𝐶𝐶 E 𝐵))
2 epelg 4950 . . . 4 (𝐶𝐴 → (𝐵 E 𝐶𝐵𝐶))
3 epelg 4950 . . . 4 (𝐵𝐴 → (𝐶 E 𝐵𝐶𝐵))
42, 3bi2anan9r 914 . . 3 ((𝐵𝐴𝐶𝐴) → ((𝐵 E 𝐶𝐶 E 𝐵) ↔ (𝐵𝐶𝐶𝐵)))
54adantl 481 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 E 𝐶𝐶 E 𝐵) ↔ (𝐵𝐶𝐶𝐵)))
61, 5mtbid 313 1 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝐶𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   class class class wbr 4583   E cep 4947   Fr wfr 4994 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-fr 4997 This theorem is referenced by:  en2lp  8393
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