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Theorem po3nr 5014
Description: A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
po3nr ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))

Proof of Theorem po3nr
StepHypRef Expression
1 po2nr 5013 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐷𝐷𝑅𝐵))
213adantr2 1219 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐷𝐷𝑅𝐵))
3 df-3an 1038 . . 3 ((𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵) ↔ ((𝐵𝑅𝐶𝐶𝑅𝐷) ∧ 𝐷𝑅𝐵))
4 potr 5012 . . . 4 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
54anim1d 587 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (((𝐵𝑅𝐶𝐶𝑅𝐷) ∧ 𝐷𝑅𝐵) → (𝐵𝑅𝐷𝐷𝑅𝐵)))
63, 5syl5bi 232 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵) → (𝐵𝑅𝐷𝐷𝑅𝐵)))
72, 6mtod 189 1 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036  wcel 1987   class class class wbr 4618   Po wpo 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-po 5000
This theorem is referenced by:  so3nr  5025
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