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Mirrors > Home > MPE Home > Th. List > tsrlin | Structured version Visualization version GIF version |
Description: A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
istsr.1 | ⊢ 𝑋 = dom 𝑅 |
Ref | Expression |
---|---|
tsrlin | ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istsr.1 | . . . . 5 ⊢ 𝑋 = dom 𝑅 | |
2 | 1 | istsr2 17816 | . . . 4 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) |
3 | 2 | simprbi 497 | . . 3 ⊢ (𝑅 ∈ TosetRel → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) |
4 | breq1 5060 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦)) | |
5 | breq2 5061 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴)) | |
6 | 4, 5 | orbi12d 912 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝐴𝑅𝑦 ∨ 𝑦𝑅𝐴))) |
7 | breq2 5061 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵)) | |
8 | breq1 5060 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴)) | |
9 | 7, 8 | orbi12d 912 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ∨ 𝑦𝑅𝐴) ↔ (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴))) |
10 | 6, 9 | rspc2v 3630 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴))) |
11 | 3, 10 | syl5com 31 | . 2 ⊢ (𝑅 ∈ TosetRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴))) |
12 | 11 | 3impib 1108 | 1 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 class class class wbr 5057 dom cdm 5548 PosetRelcps 17796 TosetRel ctsr 17797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-tsr 17799 |
This theorem is referenced by: tsrlemax 17818 ordtrest2lem 21739 ordthauslem 21919 ordthaus 21920 |
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