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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 5030 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦)) | |
2 | eqeq1 2742 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝑦 ↔ 𝐵 = 𝑦)) | |
3 | breq2 5031 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐵)) | |
4 | 1, 2, 3 | 3orbi123d 1436 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵))) |
5 | 4 | imbi2d 344 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝑅 Or 𝐴 → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵)))) |
6 | breq2 5031 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶)) | |
7 | eqeq2 2750 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐶)) | |
8 | breq1 5030 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝑦𝑅𝐵 ↔ 𝐶𝑅𝐵)) | |
9 | 6, 7, 8 | 3orbi123d 1436 | . . . 4 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))) |
10 | 9 | imbi2d 344 | . . 3 ⊢ (𝑦 = 𝐶 → ((𝑅 Or 𝐴 → (𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵)) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))) |
11 | df-so 5439 | . . . . 5 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
12 | rsp2 3124 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
13 | 11, 12 | simplbiim 508 | . . . 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 3478 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝑅 Or 𝐴 → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))) |
16 | 15 | impcom 411 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ w3o 1087 = wceq 1542 ∈ wcel 2113 ∀wral 3053 class class class wbr 5027 Po wpo 5436 Or wor 5437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-v 3399 df-un 3846 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-so 5439 |
This theorem is referenced by: sotric 5465 sotrieq 5466 somo 5474 wecmpep 5511 sorpssi 7467 soxp 7842 wfrlem10 7986 infsupprpr 9034 wemaplem2 9077 fpwwe2lem11 10134 fpwwe2lem12 10135 lttri4 10796 xmullem 12733 xmulasslem 12754 orngsqr 31072 noresle 33533 nosupbnd1lem6 33549 noinfbnd1lem6 33564 sltlin 33585 fin2so 35376 fnwe2lem3 40433 prproropf1olem4 44476 |
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