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Theorem efrn2lp 5616
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 5612 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵 E 𝐶𝐶 E 𝐵))
2 epelg 5539 . . . 4 (𝐶𝐴 → (𝐵 E 𝐶𝐵𝐶))
3 epelg 5539 . . . 4 (𝐵𝐴 → (𝐶 E 𝐵𝐶𝐵))
42, 3bi2anan9r 639 . . 3 ((𝐵𝐴𝐶𝐴) → ((𝐵 E 𝐶𝐶 E 𝐵) ↔ (𝐵𝐶𝐶𝐵)))
54adantl 483 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 E 𝐶𝐶 E 𝐵) ↔ (𝐵𝐶𝐶𝐵)))
61, 5mtbid 324 1 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝐶𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wcel 2107   class class class wbr 5106   E cep 5537   Fr wfr 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-eprel 5538  df-fr 5589
This theorem is referenced by:  en2lp  9547
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