![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
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 5602 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) | |
2 | vex 3465 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2, 2 | brcnv 5885 | . . 3 ⊢ (𝑥◡𝑅𝑥 ↔ 𝑥𝑅𝑥) |
4 | 1, 3 | sylnibr 328 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥◡𝑅𝑥) |
5 | 3anrev 1098 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) | |
6 | potr 5603 | . . . 4 ⊢ ((𝑅 Po 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥)) | |
7 | 5, 6 | sylan2b 592 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥)) |
8 | vex 3465 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 2, 8 | brcnv 5885 | . . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
10 | vex 3465 | . . . . 5 ⊢ 𝑧 ∈ V | |
11 | 8, 10 | brcnv 5885 | . . . 4 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦) |
12 | 9, 11 | anbi12ci 627 | . . 3 ⊢ ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) ↔ (𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥)) |
13 | 2, 10 | brcnv 5885 | . . 3 ⊢ (𝑥◡𝑅𝑧 ↔ 𝑧𝑅𝑥) |
14 | 7, 12, 13 | 3imtr4g 295 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) → 𝑥◡𝑅𝑧)) |
15 | 4, 14 | ispod 5599 | 1 ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5149 Po wpo 5588 ◡ccnv 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-po 5590 df-cnv 5686 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |