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Theorem epne3 7712
Description: A well-founded class contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
epne3 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝐶𝐶𝐷𝐷𝐵))

Proof of Theorem epne3
StepHypRef Expression
1 fr3nr 7711 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵 E 𝐶𝐶 E 𝐷𝐷 E 𝐵))
2 epelg 5543 . . . . 5 (𝐶𝐴 → (𝐵 E 𝐶𝐵𝐶))
323ad2ant2 1134 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐵 E 𝐶𝐵𝐶))
4 epelg 5543 . . . . 5 (𝐷𝐴 → (𝐶 E 𝐷𝐶𝐷))
543ad2ant3 1135 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐶 E 𝐷𝐶𝐷))
6 epelg 5543 . . . . 5 (𝐵𝐴 → (𝐷 E 𝐵𝐷𝐵))
763ad2ant1 1133 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐷 E 𝐵𝐷𝐵))
83, 5, 73anbi123d 1436 . . 3 ((𝐵𝐴𝐶𝐴𝐷𝐴) → ((𝐵 E 𝐶𝐶 E 𝐷𝐷 E 𝐵) ↔ (𝐵𝐶𝐶𝐷𝐷𝐵)))
98adantl 482 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵 E 𝐶𝐶 E 𝐷𝐷 E 𝐵) ↔ (𝐵𝐶𝐶𝐷𝐷𝐵)))
101, 9mtbid 323 1 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝐶𝐶𝐷𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087  wcel 2106   class class class wbr 5110   E cep 5541   Fr wfr 5590
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-eprel 5542  df-fr 5593
This theorem is referenced by: (None)
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