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Mirrors > Home > MPE Home > Th. List > Mathboxes > pocnv | Structured version Visualization version GIF version |
Description: The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
Ref | Expression |
---|---|
pocnv | ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | poirr 5075 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) | |
2 | vex 3234 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2, 2 | brcnv 5337 | . . 3 ⊢ (𝑥◡𝑅𝑥 ↔ 𝑥𝑅𝑥) |
4 | 1, 3 | sylnibr 318 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥◡𝑅𝑥) |
5 | 3anrev 1067 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) | |
6 | potr 5076 | . . . 4 ⊢ ((𝑅 Po 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥)) | |
7 | 5, 6 | sylan2b 491 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥)) |
8 | vex 3234 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 2, 8 | brcnv 5337 | . . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
10 | vex 3234 | . . . . 5 ⊢ 𝑧 ∈ V | |
11 | 8, 10 | brcnv 5337 | . . . 4 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦) |
12 | 9, 11 | anbi12ci 734 | . . 3 ⊢ ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) ↔ (𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥)) |
13 | 2, 10 | brcnv 5337 | . . 3 ⊢ (𝑥◡𝑅𝑧 ↔ 𝑧𝑅𝑥) |
14 | 7, 12, 13 | 3imtr4g 285 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) → 𝑥◡𝑅𝑧)) |
15 | 4, 14 | ispod 5072 | 1 ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 ∈ wcel 2030 class class class wbr 4685 Po wpo 5062 ◡ccnv 5142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-po 5064 df-cnv 5151 |
This theorem is referenced by: (None) |
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