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Theorem po3nr 5201
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 5200 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐷𝐷𝑅𝐵))
213adantr2 1176 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐷𝐷𝑅𝐵))
3 df-3an 1074 . . 3 ((𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵) ↔ ((𝐵𝑅𝐶𝐶𝑅𝐷) ∧ 𝐷𝑅𝐵))
4 potr 5199 . . . 4 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
54anim1d 589 . . 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 383  w3a 1072  wcel 2139   class class class wbr 4804   Po wpo 5185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-po 5187
This theorem is referenced by:  so3nr  5212
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