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Theorem ltntri 7897
 Description: Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, . Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)
Assertion
Ref Expression
ltntri

Proof of Theorem ltntri
StepHypRef Expression
1 simpll 518 . . . 4
2 simplr 519 . . . 4
3 simpr3 989 . . . 4
41, 2, 3nltled 7890 . . 3
5 simpr1 987 . . . 4
62, 1, 5nltled 7890 . . 3
71, 2letri3d 7886 . . 3
84, 6, 7mpbir2and 928 . 2
9 simpr2 988 . 2
108, 9pm2.65da 650 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   w3a 962   wceq 1331   wcel 1480   class class class wbr 3929  cr 7626   clt 7807   cle 7808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-pre-ltirr 7739  ax-pre-apti 7742 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813 This theorem is referenced by: (None)
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