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Theorem efrn2lp 5248
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 5244 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵 E 𝐶𝐶 E 𝐵))
2 epelg 5180 . . . 4 (𝐶𝐴 → (𝐵 E 𝐶𝐵𝐶))
3 epelg 5180 . . . 4 (𝐵𝐴 → (𝐶 E 𝐵𝐶𝐵))
42, 3bi2anan9r 954 . . 3 ((𝐵𝐴𝐶𝐴) → ((𝐵 E 𝐶𝐶 E 𝐵) ↔ (𝐵𝐶𝐶𝐵)))
54adantl 473 . 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 196  wa 383  wcel 2139   class class class wbr 4804   E cep 5178   Fr wfr 5222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-eprel 5179  df-fr 5225
This theorem is referenced by:  en2lp  8675
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