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Theorem son2lpi 6151
Description: A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
soi.1 𝑅 Or 𝑆
soi.2 𝑅 ⊆ (𝑆 × 𝑆)
Assertion
Ref Expression
son2lpi ¬ (𝐴𝑅𝐵𝐵𝑅𝐴)

Proof of Theorem son2lpi
StepHypRef Expression
1 soi.1 . . 3 𝑅 Or 𝑆
2 soi.2 . . 3 𝑅 ⊆ (𝑆 × 𝑆)
31, 2soirri 6149 . 2 ¬ 𝐴𝑅𝐴
41, 2sotri 6150 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴𝑅𝐴)
53, 4mto 197 1 ¬ (𝐴𝑅𝐵𝐵𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wss 3963   class class class wbr 5148   Or wor 5596   × cxp 5687
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  ax-sep 5302  ax-nul 5312  ax-pr 5438
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-rex 3069  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-opab 5211  df-po 5597  df-so 5598  df-xp 5695
This theorem is referenced by: (None)
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