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Theorem son2lpiOLD 5092
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 5085 . 2  |-  -.  A R A
41, 2sotri 5086 . 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 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    Or wor 4329    X. cxp 4703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-po 4330  df-so 4331  df-xp 4711
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