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Mirrors > Home > ILE Home > Th. List > sotricim | GIF version |
Description: One direction of sotritric 4216 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 4209 | . . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | |
2 | 1 | adantrr 470 | . . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ 𝐵𝑅𝐵) |
3 | 2 | 3adant3 986 | . . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵) |
4 | breq2 3903 | . . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐵𝑅𝐶)) | |
5 | 4 | biimprcd 159 | . . . . . 6 ⊢ (𝐵𝑅𝐶 → (𝐵 = 𝐶 → 𝐵𝑅𝐵)) |
6 | 5 | 3ad2ant3 989 | . . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝐵𝑅𝐶) → (𝐵 = 𝐶 → 𝐵𝑅𝐵)) |
7 | 3, 6 | mtod 637 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝐵𝑅𝐶) → ¬ 𝐵 = 𝐶) |
8 | 7 | 3expia 1168 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 → ¬ 𝐵 = 𝐶)) |
9 | so2nr 4213 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | |
10 | imnan 664 | . . . 4 ⊢ ((𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | |
11 | 9, 10 | sylibr 133 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵)) |
12 | 8, 11 | jcad 305 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 → (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶𝑅𝐵))) |
13 | ioran 726 | . 2 ⊢ (¬ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶𝑅𝐵)) | |
14 | 12, 13 | syl6ibr 161 | 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 103 ∨ wo 682 ∧ w3a 947 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 Or wor 4187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-po 4188 df-iso 4189 |
This theorem is referenced by: sotritric 4216 |
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