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Theorem so2nr 4104
Description: A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)
Assertion
Ref Expression
so2nr ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))

Proof of Theorem so2nr
StepHypRef Expression
1 sopo 4096 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 po2nr 4092 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
31, 2sylan 277 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wcel 1434   class class class wbr 3805   Po wpo 4077   Or wor 4078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2611  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-po 4079  df-iso 4080
This theorem is referenced by:  sotricim  4106  cauappcvgprlemdisj  6938  cauappcvgprlemladdru  6943  cauappcvgprlemladdrl  6944  caucvgprlemnbj  6954  caucvgprprlemnbj  6980  ltnsym2  7303
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