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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 5480 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) | |
2 | vex 3412 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2, 2 | brcnv 5751 | . . 3 ⊢ (𝑥◡𝑅𝑥 ↔ 𝑥𝑅𝑥) |
4 | 1, 3 | sylnibr 332 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥◡𝑅𝑥) |
5 | 3anrev 1103 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) | |
6 | potr 5481 | . . . 4 ⊢ ((𝑅 Po 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥)) | |
7 | 5, 6 | sylan2b 597 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥)) |
8 | vex 3412 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 2, 8 | brcnv 5751 | . . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
10 | vex 3412 | . . . . 5 ⊢ 𝑧 ∈ V | |
11 | 8, 10 | brcnv 5751 | . . . 4 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦) |
12 | 9, 11 | anbi12ci 631 | . . 3 ⊢ ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) ↔ (𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥)) |
13 | 2, 10 | brcnv 5751 | . . 3 ⊢ (𝑥◡𝑅𝑧 ↔ 𝑧𝑅𝑥) |
14 | 7, 12, 13 | 3imtr4g 299 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) → 𝑥◡𝑅𝑧)) |
15 | 4, 14 | ispod 5477 | 1 ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2110 class class class wbr 5053 Po wpo 5466 ◡ccnv 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-po 5468 df-cnv 5559 |
This theorem is referenced by: (None) |
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