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Theorem sotricim 4174
Description: One direction of sotritric 4175 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 4168 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
21adantrr 464 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ 𝐵𝑅𝐵)
323adant3 966 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
4 breq2 3871 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐵𝑅𝐶))
54biimprcd 159 . . . . . 6 (𝐵𝑅𝐶 → (𝐵 = 𝐶𝐵𝑅𝐵))
653ad2ant3 969 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → (𝐵 = 𝐶𝐵𝑅𝐵))
73, 6mtod 627 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵 = 𝐶)
873expia 1148 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ 𝐵 = 𝐶))
9 so2nr 4172 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
10 imnan 662 . . . 4 ((𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
119, 10sylibr 133 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵))
128, 11jcad 302 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶𝑅𝐵)))
13 ioran 707 . 2 (¬ (𝐵 = 𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶𝑅𝐵))
1412, 13syl6ibr 161 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 667  w3a 927   = wceq 1296  wcel 1445   class class class wbr 3867   Or wor 4146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-po 4147  df-iso 4148
This theorem is referenced by:  sotritric  4175
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