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Theorem po2nr 5604
Description: A partial order 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 5602 . . 3 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
21adantrr 715 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ 𝐵𝑅𝐵)
3 potr 5603 . . . . . 6 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))
433exp2 1351 . . . . 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 417 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))
82, 7mtod 197 1 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wcel 2098   class class class wbr 5149   Po wpo 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-po 5590
This theorem is referenced by:  po3nr  5605  so2nr  5616  soisoi  7335  poxp2  8148  poxp3  8155  wemaplem2  9577  pospo  18345  poprelb  47003
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