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Theorem ltntri 8040
Description: Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy,  A  <  B  \/  A  =  B  \/  B  <  A. 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  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )

Proof of Theorem ltntri
StepHypRef Expression
1 simpll 524 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )  ->  A  e.  RR )
2 simplr 525 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )  ->  B  e.  RR )
3 simpr3 1000 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )  ->  -.  B  <  A )
41, 2, 3nltled 8033 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )  ->  A  <_  B )
5 simpr1 998 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )  ->  -.  A  <  B )
62, 1, 5nltled 8033 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )  ->  B  <_  A )
71, 2letri3d 8028 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )  -> 
( A  =  B  <-> 
( A  <_  B  /\  B  <_  A ) ) )
84, 6, 7mpbir2and 939 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )  ->  A  =  B )
9 simpr2 999 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )  ->  -.  A  =  B
)
108, 9pm2.65da 656 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    /\ w3a 973    = wceq 1348    e. wcel 2141   class class class wbr 3987   RRcr 7766    < clt 7947    <_ cle 7948
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-pre-ltirr 7879  ax-pre-apti 7882
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-xp 4615  df-cnv 4617  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953
This theorem is referenced by: (None)
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