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Mirrors > Home > MPE Home > Th. List > solin | Structured version Visualization version GIF version |
Description: A strict order relation is linear (satisfies trichotomy). (Contributed by NM, 21-Jan-1996.) |
Ref | Expression |
---|---|
solin | ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4586 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦)) | |
2 | eqeq1 2614 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝑦 ↔ 𝐵 = 𝑦)) | |
3 | breq2 4587 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐵)) | |
4 | 1, 2, 3 | 3orbi123d 1390 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵))) |
5 | 4 | imbi2d 329 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝑅 Or 𝐴 → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵)))) |
6 | breq2 4587 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶)) | |
7 | eqeq2 2621 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐶)) | |
8 | breq1 4586 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝑦𝑅𝐵 ↔ 𝐶𝑅𝐵)) | |
9 | 6, 7, 8 | 3orbi123d 1390 | . . . 4 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))) |
10 | 9 | imbi2d 329 | . . 3 ⊢ (𝑦 = 𝐶 → ((𝑅 Or 𝐴 → (𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵)) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))) |
11 | df-so 4960 | . . . . 5 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
12 | rsp2 2920 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
13 | 11, 12 | simplbiim 657 | . . . 4 ⊢ (𝑅 Or 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) |
14 | 13 | com12 32 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑅 Or 𝐴 → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) |
15 | 5, 10, 14 | vtocl2ga 3247 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝑅 Or 𝐴 → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))) |
16 | 15 | impcom 445 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 Po wpo 4957 Or wor 4958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-so 4960 |
This theorem is referenced by: sotric 4985 sotrieq 4986 somo 4993 wecmpep 5030 sorpssi 6841 soxp 7177 wfrlem10 7311 wemaplem2 8335 fpwwe2lem12 9342 fpwwe2lem13 9343 lttri4 10001 xmullem 11966 xmulasslem 11987 orngsqr 29135 socnv 30908 slttri 31072 fin2so 32566 fnwe2lem3 36640 |
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