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Theorem so2nr 4252
 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 4244 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 po2nr 4240 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
31, 2sylan 281 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∈ wcel 1481   class class class wbr 3938   Po wpo 4225   Or wor 4226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2692  df-un 3081  df-sn 3539  df-pr 3540  df-op 3542  df-br 3939  df-po 4227  df-iso 4228 This theorem is referenced by:  sotricim  4254  cauappcvgprlemdisj  7503  cauappcvgprlemladdru  7508  cauappcvgprlemladdrl  7509  caucvgprlemnbj  7519  caucvgprprlemnbj  7545  suplocexprlemmu  7570  ltnsym2  7898
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