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Theorem son2lpiOLD 5267
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 5260 . 2  |-  -.  A R A
41, 2sotri 5261 . 2  |-  ( ( A R B  /\  B R A )  ->  A R A )
53, 4mto 169 1  |-  -.  ( A R B  /\  B R A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    e. wcel 1725   _Vcvv 2956    C_ wss 3320   class class class wbr 4212    Or wor 4502    X. cxp 4876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-po 4503  df-so 4504  df-xp 4884
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