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Theorem so2nr 5530
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 5523 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 po2nr 5518 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
31, 2sylan 580 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2110   class class class wbr 5079   Po wpo 5502   Or wor 5503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-po 5504  df-so 5505
This theorem is referenced by:  sotric  5532  somincom  6038  fisupg  9040  suppr  9208  infpr  9240  genpnnp  10762  ltnsym2  11074
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