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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 6054 | . 2 ⊢ Rel ◡(𝐴 ∖ 𝐵) | |
2 | difss 4089 | . . 3 ⊢ (◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 | |
3 | relcnv 6054 | . . 3 ⊢ Rel ◡𝐴 | |
4 | relss 5735 | . . 3 ⊢ ((◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 → (Rel ◡𝐴 → Rel (◡𝐴 ∖ ◡𝐵))) | |
5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel (◡𝐴 ∖ ◡𝐵) |
6 | eldif 3918 | . . 3 ⊢ (〈𝑦, 𝑥〉 ∈ (𝐴 ∖ 𝐵) ↔ (〈𝑦, 𝑥〉 ∈ 𝐴 ∧ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵)) | |
7 | vex 3447 | . . . 4 ⊢ 𝑥 ∈ V | |
8 | vex 3447 | . . . 4 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opelcnv 5835 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∖ 𝐵) ↔ 〈𝑦, 𝑥〉 ∈ (𝐴 ∖ 𝐵)) |
10 | eldif 3918 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝐴 ∖ ◡𝐵) ↔ (〈𝑥, 𝑦〉 ∈ ◡𝐴 ∧ ¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵)) | |
11 | 7, 8 | opelcnv 5835 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡𝐴 ↔ 〈𝑦, 𝑥〉 ∈ 𝐴) |
12 | 7, 8 | opelcnv 5835 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ ◡𝐵 ↔ 〈𝑦, 𝑥〉 ∈ 𝐵) |
13 | 12 | notbii 319 | . . . . 5 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵 ↔ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵) |
14 | 11, 13 | anbi12i 627 | . . . 4 ⊢ ((〈𝑥, 𝑦〉 ∈ ◡𝐴 ∧ ¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵) ↔ (〈𝑦, 𝑥〉 ∈ 𝐴 ∧ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵)) |
15 | 10, 14 | bitri 274 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝐴 ∖ ◡𝐵) ↔ (〈𝑦, 𝑥〉 ∈ 𝐴 ∧ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵)) |
16 | 6, 9, 15 | 3bitr4i 302 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∖ 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (◡𝐴 ∖ ◡𝐵)) |
17 | 1, 5, 16 | eqrelriiv 5744 | 1 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3905 ⊆ wss 3908 〈cop 4590 ◡ccnv 5630 Rel wrel 5636 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-xp 5637 df-rel 5638 df-cnv 5639 |
This theorem is referenced by: cnvin 6095 gtiso 31498 gsumhashmul 31781 mthmpps 34045 cnvnonrel 41802 |
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