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Theorem po2nr 5013
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 5011 . . 3 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
21adantrr 752 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ 𝐵𝑅𝐵)
3 potr 5012 . . . . . 6 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))
433exp2 1282 . . . . 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 449 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))
82, 7mtod 189 1 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wcel 1992   class class class wbr 4618   Po wpo 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-po 5000
This theorem is referenced by:  po3nr  5014  so2nr  5024  soisoi  6533  wemaplem2  8397  pospo  16889
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