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Theorem son2lpi 4748
 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 4746 . 2 ¬ 𝐴𝑅𝐴
41, 2sotri 4747 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴𝑅𝐴)
53, 4mto 598 1 ¬ (𝐴𝑅𝐵𝐵𝑅𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 101   ⊆ wss 2944   class class class wbr 3791   Or wor 4059   × cxp 4370 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-po 4060  df-iso 4061  df-xp 4378 This theorem is referenced by:  nqprdisj  6699  ltexprlemdisj  6761  recexprlemdisj  6785  caucvgprlemnkj  6821  caucvgprprlemnkltj  6844  caucvgprprlemnkeqj  6845  caucvgprprlemnjltk  6846
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