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Theorem po3nr 5558
Description: A partial order 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 5557 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐷𝐷𝑅𝐵))
213adantr2 1170 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐷𝐷𝑅𝐵))
3 df-3an 1089 . . 3 ((𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵) ↔ ((𝐵𝑅𝐶𝐶𝑅𝐷) ∧ 𝐷𝑅𝐵))
4 potr 5556 . . . 4 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
54anim1d 611 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (((𝐵𝑅𝐶𝐶𝑅𝐷) ∧ 𝐷𝑅𝐵) → (𝐵𝑅𝐷𝐷𝑅𝐵)))
63, 5biimtrid 241 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵) → (𝐵𝑅𝐷𝐷𝑅𝐵)))
72, 6mtod 197 1 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087  wcel 2106   class class class wbr 5103   Po wpo 5541
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 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-po 5543
This theorem is referenced by:  so3nr  5570
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