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Theorem po3nr 4038
Description: A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
po3nr ((𝑅 Po A (B A 𝐶 A 𝐷 A)) → ¬ (B𝑅𝐶 𝐶𝑅𝐷 𝐷𝑅B))

Proof of Theorem po3nr
StepHypRef Expression
1 po2nr 4037 . . 3 ((𝑅 Po A (B A 𝐷 A)) → ¬ (B𝑅𝐷 𝐷𝑅B))
213adantr2 1063 . 2 ((𝑅 Po A (B A 𝐶 A 𝐷 A)) → ¬ (B𝑅𝐷 𝐷𝑅B))
3 df-3an 886 . . 3 ((B𝑅𝐶 𝐶𝑅𝐷 𝐷𝑅B) ↔ ((B𝑅𝐶 𝐶𝑅𝐷) 𝐷𝑅B))
4 potr 4036 . . . 4 ((𝑅 Po A (B A 𝐶 A 𝐷 A)) → ((B𝑅𝐶 𝐶𝑅𝐷) → B𝑅𝐷))
54anim1d 319 . . 3 ((𝑅 Po A (B A 𝐶 A 𝐷 A)) → (((B𝑅𝐶 𝐶𝑅𝐷) 𝐷𝑅B) → (B𝑅𝐷 𝐷𝑅B)))
63, 5syl5bi 141 . 2 ((𝑅 Po A (B A 𝐶 A 𝐷 A)) → ((B𝑅𝐶 𝐶𝑅𝐷 𝐷𝑅B) → (B𝑅𝐷 𝐷𝑅B)))
72, 6mtod 588 1 ((𝑅 Po A (B A 𝐶 A 𝐷 A)) → ¬ (B𝑅𝐶 𝐶𝑅𝐷 𝐷𝑅B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   w3a 884   wcel 1390   class class class wbr 3755   Po wpo 4022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-po 4024
This theorem is referenced by:  so3nr  4050
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