ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltntri GIF version

Theorem ltntri 8087
Description: Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, 𝐴 < 𝐵𝐴 = 𝐵𝐵 < 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 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴))

Proof of Theorem ltntri
StepHypRef Expression
1 simpll 527 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴)) → 𝐴 ∈ ℝ)
2 simplr 528 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴)) → 𝐵 ∈ ℝ)
3 simpr3 1005 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴)) → ¬ 𝐵 < 𝐴)
41, 2, 3nltled 8080 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴)) → 𝐴𝐵)
5 simpr1 1003 . . . 4 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴)) → ¬ 𝐴 < 𝐵)
62, 1, 5nltled 8080 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴)) → 𝐵𝐴)
71, 2letri3d 8075 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴)) → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
84, 6, 7mpbir2and 944 . 2 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴)) → 𝐴 = 𝐵)
9 simpr2 1004 . 2 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴)) → ¬ 𝐴 = 𝐵)
108, 9pm2.65da 661 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148   class class class wbr 4005  cr 7812   < clt 7994  cle 7995
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-pre-ltirr 7925  ax-pre-apti 7928
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator