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Theorem epne3 7212
Description: A set well-founded by epsilon 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 7211 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵 E 𝐶𝐶 E 𝐷𝐷 E 𝐵))
2 epelg 5224 . . . . 5 (𝐶𝐴 → (𝐵 E 𝐶𝐵𝐶))
323ad2ant2 1165 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐵 E 𝐶𝐵𝐶))
4 epelg 5224 . . . . 5 (𝐷𝐴 → (𝐶 E 𝐷𝐶𝐷))
543ad2ant3 1166 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐶 E 𝐷𝐶𝐷))
6 epelg 5224 . . . . 5 (𝐵𝐴 → (𝐷 E 𝐵𝐷𝐵))
763ad2ant1 1164 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐷 E 𝐵𝐷𝐵))
83, 5, 73anbi123d 1561 . . 3 ((𝐵𝐴𝐶𝐴𝐷𝐴) → ((𝐵 E 𝐶𝐶 E 𝐷𝐷 E 𝐵) ↔ (𝐵𝐶𝐶𝐷𝐷𝐵)))
98adantl 474 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵 E 𝐶𝐶 E 𝐷𝐷 E 𝐵) ↔ (𝐵𝐶𝐶𝐷𝐷𝐵)))
101, 9mtbid 316 1 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝐶𝐶𝐷𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  w3a 1108  wcel 2157   class class class wbr 4841   E cep 5222   Fr wfr 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-eprel 5223  df-fr 5269
This theorem is referenced by: (None)
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