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Theorem son2lpi 5037
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 5035 . 2  |-  -.  A R A
41, 2sotri 5036 . 2  |-  ( ( A R B  /\  B R A )  ->  A R A )
53, 4mto 663 1  |-  -.  ( A R B  /\  B R A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    C_ wss 3141   class class class wbr 4015    Or wor 4307    X. cxp 4636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-po 4308  df-iso 4309  df-xp 4644
This theorem is referenced by:  nqprdisj  7556  ltexprlemdisj  7618  recexprlemdisj  7642  caucvgprlemnkj  7678  caucvgprprlemnkltj  7701  caucvgprprlemnkeqj  7702  caucvgprprlemnjltk  7703
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