![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnvdif | Structured version Visualization version GIF version |
Description: Distributive law for converse over class difference. (Contributed by Mario Carneiro, 26-Jun-2014.) |
Ref | Expression |
---|---|
cnvdif | ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6124 | . 2 ⊢ Rel ◡(𝐴 ∖ 𝐵) | |
2 | difss 4145 | . . 3 ⊢ (◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 | |
3 | relcnv 6124 | . . 3 ⊢ Rel ◡𝐴 | |
4 | relss 5793 | . . 3 ⊢ ((◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 → (Rel ◡𝐴 → Rel (◡𝐴 ∖ ◡𝐵))) | |
5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel (◡𝐴 ∖ ◡𝐵) |
6 | eldif 3972 | . . 3 ⊢ (〈𝑦, 𝑥〉 ∈ (𝐴 ∖ 𝐵) ↔ (〈𝑦, 𝑥〉 ∈ 𝐴 ∧ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵)) | |
7 | vex 3481 | . . . 4 ⊢ 𝑥 ∈ V | |
8 | vex 3481 | . . . 4 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opelcnv 5894 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∖ 𝐵) ↔ 〈𝑦, 𝑥〉 ∈ (𝐴 ∖ 𝐵)) |
10 | eldif 3972 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝐴 ∖ ◡𝐵) ↔ (〈𝑥, 𝑦〉 ∈ ◡𝐴 ∧ ¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵)) | |
11 | 7, 8 | opelcnv 5894 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡𝐴 ↔ 〈𝑦, 𝑥〉 ∈ 𝐴) |
12 | 7, 8 | opelcnv 5894 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ ◡𝐵 ↔ 〈𝑦, 𝑥〉 ∈ 𝐵) |
13 | 12 | notbii 320 | . . . . 5 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵 ↔ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵) |
14 | 11, 13 | anbi12i 628 | . . . 4 ⊢ ((〈𝑥, 𝑦〉 ∈ ◡𝐴 ∧ ¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵) ↔ (〈𝑦, 𝑥〉 ∈ 𝐴 ∧ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵)) |
15 | 10, 14 | bitri 275 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝐴 ∖ ◡𝐵) ↔ (〈𝑦, 𝑥〉 ∈ 𝐴 ∧ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵)) |
16 | 6, 9, 15 | 3bitr4i 303 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∖ 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (◡𝐴 ∖ ◡𝐵)) |
17 | 1, 5, 16 | eqrelriiv 5802 | 1 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∖ cdif 3959 ⊆ wss 3962 〈cop 4636 ◡ccnv 5687 Rel wrel 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 |
This theorem is referenced by: cnvin 6166 gtiso 32715 gsumhashmul 33046 mthmpps 35566 cnvnonrel 43577 |
Copyright terms: Public domain | W3C validator |