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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 5449 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) | |
2 | vex 3444 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2, 2 | brcnv 5717 | . . 3 ⊢ (𝑥◡𝑅𝑥 ↔ 𝑥𝑅𝑥) |
4 | 1, 3 | sylnibr 332 | . 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 5450 | . . . 4 ⊢ ((𝑅 Po 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥)) | |
7 | 5, 6 | sylan2b 596 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑧𝑅𝑥)) |
8 | vex 3444 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 2, 8 | brcnv 5717 | . . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
10 | vex 3444 | . . . . 5 ⊢ 𝑧 ∈ V | |
11 | 8, 10 | brcnv 5717 | . . . 4 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦) |
12 | 9, 11 | anbi12ci 630 | . . 3 ⊢ ((𝑥◡𝑅𝑦 ∧ 𝑦◡𝑅𝑧) ↔ (𝑧𝑅𝑦 ∧ 𝑦𝑅𝑥)) |
13 | 2, 10 | brcnv 5717 | . . 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 5446 | 1 ⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 class class class wbr 5030 Po wpo 5436 ◡ccnv 5518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-po 5438 df-cnv 5527 |
This theorem is referenced by: (None) |
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