MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  po3nr Structured version   Visualization version   GIF version

Theorem po3nr 5606
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 5605 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐷𝐷𝑅𝐵))
213adantr2 1170 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐷𝐷𝑅𝐵))
3 df-3an 1088 . . 3 ((𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵) ↔ ((𝐵𝑅𝐶𝐶𝑅𝐷) ∧ 𝐷𝑅𝐵))
4 potr 5604 . . . 4 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
54anim1d 611 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (((𝐵𝑅𝐶𝐶𝑅𝐷) ∧ 𝐷𝑅𝐵) → (𝐵𝑅𝐷𝐷𝑅𝐵)))
63, 5biimtrid 242 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵) → (𝐵𝑅𝐷𝐷𝑅𝐵)))
72, 6mtod 198 1 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086  wcel 2107   class class class wbr 5142   Po wpo 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-po 5591
This theorem is referenced by:  so3nr  5620
  Copyright terms: Public domain W3C validator