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Theorem epne3 7762
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 7761 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵 E 𝐶𝐶 E 𝐷𝐷 E 𝐵))
2 epelg 5580 . . . . 5 (𝐶𝐴 → (𝐵 E 𝐶𝐵𝐶))
323ad2ant2 1132 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐵 E 𝐶𝐵𝐶))
4 epelg 5580 . . . . 5 (𝐷𝐴 → (𝐶 E 𝐷𝐶𝐷))
543ad2ant3 1133 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐶 E 𝐷𝐶𝐷))
6 epelg 5580 . . . . 5 (𝐵𝐴 → (𝐷 E 𝐵𝐷𝐵))
763ad2ant1 1131 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐷 E 𝐵𝐷𝐵))
83, 5, 73anbi123d 1434 . . 3 ((𝐵𝐴𝐶𝐴𝐷𝐴) → ((𝐵 E 𝐶𝐶 E 𝐷𝐷 E 𝐵) ↔ (𝐵𝐶𝐶𝐷𝐷𝐵)))
98adantl 480 . 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 394  w3a 1085  wcel 2104   class class class wbr 5147   E cep 5578   Fr wfr 5627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-eprel 5579  df-fr 5630
This theorem is referenced by: (None)
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