MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  so2nr Structured version   Visualization version   GIF version

Theorem so2nr 5624
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 5616 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 po2nr 5611 . 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 395  wcel 2106   class class class wbr 5148   Po wpo 5595   Or wor 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-po 5597  df-so 5598
This theorem is referenced by:  sotric  5626  somincom  6157  fisupg  9322  suppr  9509  infpr  9541  genpnnp  11043  ltnsym2  11358
  Copyright terms: Public domain W3C validator