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Theorem son2lpi 4664
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 ¬ (A𝑅B B𝑅A)

Proof of Theorem son2lpi
StepHypRef Expression
1 soi.1 . . 3 𝑅 Or 𝑆
2 soi.2 . . 3 𝑅 ⊆ (𝑆 × 𝑆)
31, 2soirri 4662 . 2 ¬ A𝑅A
41, 2sotri 4663 . 2 ((A𝑅B B𝑅A) → A𝑅A)
53, 4mto 587 1 ¬ (A𝑅B B𝑅A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97  wss 2911   class class class wbr 3755   Or wor 4023   × cxp 4286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-po 4024  df-iso 4025  df-xp 4294
This theorem is referenced by:  nqprdisj  6527  ltexprlemdisj  6580  recexprlemdisj  6602  caucvgprlemnkj  6637
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