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Theorem son2lpi 5559
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 5557 . 2 ¬ 𝐴𝑅𝐴
41, 2sotri 5558 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴𝑅𝐴)
53, 4mto 188 1 ¬ (𝐴𝑅𝐵𝐵𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383  wss 3607   class class class wbr 4685   Or wor 5063   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-xp 5149
This theorem is referenced by: (None)
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