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Mirrors > Home > ILE Home > Th. List > sotricim | GIF version |
Description: One direction of sotritric 4052 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.) |
Ref | Expression |
---|---|
sotricim | ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → (B𝑅𝐶 → ¬ (B = 𝐶 ∨ 𝐶𝑅B))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sonr 4045 | . . . . . . 7 ⊢ ((𝑅 Or A ∧ B ∈ A) → ¬ B𝑅B) | |
2 | 1 | adantrr 448 | . . . . . 6 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → ¬ B𝑅B) |
3 | 2 | 3adant3 923 | . . . . 5 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A) ∧ B𝑅𝐶) → ¬ B𝑅B) |
4 | breq2 3759 | . . . . . . 7 ⊢ (B = 𝐶 → (B𝑅B ↔ B𝑅𝐶)) | |
5 | 4 | biimprcd 149 | . . . . . 6 ⊢ (B𝑅𝐶 → (B = 𝐶 → B𝑅B)) |
6 | 5 | 3ad2ant3 926 | . . . . 5 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A) ∧ B𝑅𝐶) → (B = 𝐶 → B𝑅B)) |
7 | 3, 6 | mtod 588 | . . . 4 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A) ∧ B𝑅𝐶) → ¬ B = 𝐶) |
8 | 7 | 3expia 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)) | |
11 | 9, 10 | sylibr 137 | . . 3 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → (B𝑅𝐶 → ¬ 𝐶𝑅B)) |
12 | 8, 11 | jcad 291 | . 2 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → (B𝑅𝐶 → (¬ B = 𝐶 ∧ ¬ 𝐶𝑅B))) |
13 | ioran 668 | . 2 ⊢ (¬ (B = 𝐶 ∨ 𝐶𝑅B) ↔ (¬ B = 𝐶 ∧ ¬ 𝐶𝑅B)) | |
14 | 12, 13 | syl6ibr 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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