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Theorem son2lpi 5265
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  C_  ( S  X.  S
)
Assertion
Ref Expression
son2lpi  |-  -.  ( A R B  /\  B R A )

Proof of Theorem son2lpi
StepHypRef Expression
1 soi.1 . . 3  |-  R  Or  S
2 soi.2 . . 3  |-  R  C_  ( S  X.  S
)
31, 2soirri 5263 . 2  |-  -.  A R A
41, 2sotri 5264 . 2  |-  ( ( A R B  /\  B R A )  ->  A R A )
53, 4mto 170 1  |-  -.  ( A R B  /\  B R A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    C_ wss 3322   class class class wbr 4215    Or wor 4505    X. cxp 4879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-po 4506  df-so 4507  df-xp 4887
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