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

Theorem sotricim 4114
Description: One direction of sotritric 4115 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.)
Assertion
Ref Expression
sotricim ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))

Proof of Theorem sotricim
StepHypRef Expression
1 sonr 4108 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
21adantrr 463 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ 𝐵𝑅𝐵)
323adant3 959 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
4 breq2 3815 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐵𝑅𝐶))
54biimprcd 158 . . . . . 6 (𝐵𝑅𝐶 → (𝐵 = 𝐶𝐵𝑅𝐵))
653ad2ant3 962 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → (𝐵 = 𝐶𝐵𝑅𝐵))
73, 6mtod 622 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵 = 𝐶)
873expia 1141 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ 𝐵 = 𝐶))
9 so2nr 4112 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
10 imnan 657 . . . 4 ((𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
119, 10sylibr 132 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵))
128, 11jcad 301 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶𝑅𝐵)))
13 ioran 702 . 2 (¬ (𝐵 = 𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶𝑅𝐵))
1412, 13syl6ibr 160 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  w3a 920   = wceq 1285  wcel 1434   class class class wbr 3811   Or wor 4086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-po 4087  df-iso 4088
This theorem is referenced by:  sotritric  4115
  Copyright terms: Public domain W3C validator