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Theorem po3nr 4075
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 4074 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐷𝐷𝑅𝐵))
213adantr2 1075 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐷𝐷𝑅𝐵))
3 df-3an 898 . . 3 ((𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵) ↔ ((𝐵𝑅𝐶𝐶𝑅𝐷) ∧ 𝐷𝑅𝐵))
4 potr 4073 . . . 4 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
54anim1d 323 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (((𝐵𝑅𝐶𝐶𝑅𝐷) ∧ 𝐷𝑅𝐵) → (𝐵𝑅𝐷𝐷𝑅𝐵)))
63, 5syl5bi 145 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵) → (𝐵𝑅𝐷𝐷𝑅𝐵)))
72, 6mtod 599 1 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  w3a 896  wcel 1409   class class class wbr 3792   Po wpo 4059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-po 4061
This theorem is referenced by:  so3nr  4087
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