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Theorem son2lpiOLD 5076
Description: A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
soiOLD.1  |-  A  e. 
_V
soiOLD.2  |-  R  Or  S
soiOLD.3  |-  R  C_  ( S  X.  S
)
son2lpiOLD.4  |-  B  e. 
_V
Assertion
Ref Expression
son2lpiOLD  |-  -.  ( A R B  /\  B R A )

Proof of Theorem son2lpiOLD
StepHypRef Expression
1 soiOLD.2 . . 3  |-  R  Or  S
2 soiOLD.3 . . 3  |-  R  C_  ( S  X.  S
)
31, 2soirri 5069 . 2  |-  -.  A R A
41, 2sotri 5070 . 2  |-  ( ( A R B  /\  B R A )  ->  A R A )
53, 4mto 167 1  |-  -.  ( A R B  /\  B R A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    e. wcel 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    Or wor 4313    X. cxp 4687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-xp 4695
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