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Theorem po2nr 4071
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 4069 . . 3 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
21adantrr 456 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ 𝐵𝑅𝐵)
3 potr 4070 . . . . . 6 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))
433exp2 1131 . . . . 5 (𝑅 Po 𝐴 → (𝐵𝐴 → (𝐶𝐴 → (𝐵𝐴 → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
54com34 81 . . . 4 (𝑅 Po 𝐴 → (𝐵𝐴 → (𝐵𝐴 → (𝐶𝐴 → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
65pm2.43d 48 . . 3 (𝑅 Po 𝐴 → (𝐵𝐴 → (𝐶𝐴 → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))))
76imp32 248 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))
82, 7mtod 597 1 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wcel 1407   class class class wbr 3789   Po wpo 4056
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 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-v 2574  df-un 2947  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-po 4058
This theorem is referenced by:  po3nr  4072  so2nr  4083
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