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Theorem sotricim 4051
 Description: One direction of sotritric 4052 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.)
Assertion
Ref Expression
sotricim ((𝑅 Or A (B A 𝐶 A)) → (B𝑅𝐶 → ¬ (B = 𝐶 𝐶𝑅B)))

Proof of Theorem sotricim
StepHypRef Expression
1 sonr 4045 . . . . . . 7 ((𝑅 Or A B A) → ¬ B𝑅B)
21adantrr 448 . . . . . 6 ((𝑅 Or A (B A 𝐶 A)) → ¬ B𝑅B)
323adant3 923 . . . . 5 ((𝑅 Or A (B A 𝐶 A) B𝑅𝐶) → ¬ B𝑅B)
4 breq2 3759 . . . . . . 7 (B = 𝐶 → (B𝑅BB𝑅𝐶))
54biimprcd 149 . . . . . 6 (B𝑅𝐶 → (B = 𝐶B𝑅B))
653ad2ant3 926 . . . . 5 ((𝑅 Or A (B A 𝐶 A) B𝑅𝐶) → (B = 𝐶B𝑅B))
73, 6mtod 588 . . . 4 ((𝑅 Or A (B A 𝐶 A) B𝑅𝐶) → ¬ B = 𝐶)
873expia 1105 . . 3 ((𝑅 Or A (B A 𝐶 A)) → (B𝑅𝐶 → ¬ B = 𝐶))
9 so2nr 4049 . . . 4 ((𝑅 Or A (B A 𝐶 A)) → ¬ (B𝑅𝐶 𝐶𝑅B))
10 imnan 623 . . . 4 ((B𝑅𝐶 → ¬ 𝐶𝑅B) ↔ ¬ (B𝑅𝐶 𝐶𝑅B))
119, 10sylibr 137 . . 3 ((𝑅 Or A (B A 𝐶 A)) → (B𝑅𝐶 → ¬ 𝐶𝑅B))
128, 11jcad 291 . 2 ((𝑅 Or A (B A 𝐶 A)) → (B𝑅𝐶 → (¬ B = 𝐶 ¬ 𝐶𝑅B)))
13 ioran 668 . 2 (¬ (B = 𝐶 𝐶𝑅B) ↔ (¬ B = 𝐶 ¬ 𝐶𝑅B))
1412, 13syl6ibr 151 1 ((𝑅 Or A (B A 𝐶 A)) → (B𝑅𝐶 → ¬ (B = 𝐶 𝐶𝑅B)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 628   ∧ w3a 884   = wceq 1242   ∈ wcel 1390   class class class wbr 3755   Or wor 4023 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-po 4024  df-iso 4025 This theorem is referenced by:  sotritric  4052
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