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Theorem po2nr 5382
Description: A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
po2nr ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))

Proof of Theorem po2nr
StepHypRef Expression
1 poirr 5380 . . 3 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
21adantrr 713 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ 𝐵𝑅𝐵)
3 potr 5381 . . . . . 6 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))
433exp2 1347 . . . . 5 (𝑅 Po 𝐴 → (𝐵𝐴 → (𝐶𝐴 → (𝐵𝐴 → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
54com34 91 . . . 4 (𝑅 Po 𝐴 → (𝐵𝐴 → (𝐵𝐴 → (𝐶𝐴 → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
65pm2.43d 53 . . 3 (𝑅 Po 𝐴 → (𝐵𝐴 → (𝐶𝐴 → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))))
76imp32 419 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))
82, 7mtod 199 1 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2083   class class class wbr 4968   Po wpo 5367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-po 5369
This theorem is referenced by:  po3nr  5383  so2nr  5394  soisoi  6951  wemaplem2  8864  pospo  17416  poprelb  43190
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