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Theorem son2lpi 4975
 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 4973 . 2 ¬ 𝐴𝑅𝐴
41, 2sotri 4974 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴𝑅𝐴)
53, 4mto 652 1 ¬ (𝐴𝑅𝐵𝐵𝑅𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 103   ⊆ wss 3098   class class class wbr 3961   Or wor 4250   × cxp 4577 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-po 4251  df-iso 4252  df-xp 4585 This theorem is referenced by:  nqprdisj  7443  ltexprlemdisj  7505  recexprlemdisj  7529  caucvgprlemnkj  7565  caucvgprprlemnkltj  7588  caucvgprprlemnkeqj  7589  caucvgprprlemnjltk  7590
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