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Theorem efrn2lp 5505
Description: A well-founded class 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 5501 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵 E 𝐶𝐶 E 𝐵))
2 epelg 5434 . . . 4 (𝐶𝐴 → (𝐵 E 𝐶𝐵𝐶))
3 epelg 5434 . . . 4 (𝐵𝐴 → (𝐶 E 𝐵𝐶𝐵))
42, 3bi2anan9r 639 . . 3 ((𝐵𝐴𝐶𝐴) → ((𝐵 E 𝐶𝐶 E 𝐵) ↔ (𝐵𝐶𝐶𝐵)))
54adantl 485 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 E 𝐶𝐶 E 𝐵) ↔ (𝐵𝐶𝐶𝐵)))
61, 5mtbid 327 1 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝐶𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2112   class class class wbr 5033   E cep 5432   Fr wfr 5479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-eprel 5433  df-fr 5482
This theorem is referenced by:  en2lp  9057
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