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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 6122 | . 2 ⊢ Rel ◡(𝐴 ∖ 𝐵) | |
| 2 | difss 4136 | . . 3 ⊢ (◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 | |
| 3 | relcnv 6122 | . . 3 ⊢ Rel ◡𝐴 | |
| 4 | relss 5791 | . . 3 ⊢ ((◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 → (Rel ◡𝐴 → Rel (◡𝐴 ∖ ◡𝐵))) | |
| 5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel (◡𝐴 ∖ ◡𝐵) |
| 6 | eldif 3961 | . . 3 ⊢ (〈𝑦, 𝑥〉 ∈ (𝐴 ∖ 𝐵) ↔ (〈𝑦, 𝑥〉 ∈ 𝐴 ∧ ¬ 〈𝑦, 𝑥〉 ∈ 𝐵)) | |
| 7 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 8 | vex 3484 | . . . 4 ⊢ 𝑦 ∈ V | |
| 9 | 7, 8 | opelcnv 5892 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(𝐴 ∖ 𝐵) ↔ 〈𝑦, 𝑥〉 ∈ (𝐴 ∖ 𝐵)) |
| 10 | eldif 3961 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝐴 ∖ ◡𝐵) ↔ (〈𝑥, 𝑦〉 ∈ ◡𝐴 ∧ ¬ 〈𝑥, 𝑦〉 ∈ ◡𝐵)) | |
| 11 | 7, 8 | opelcnv 5892 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡𝐴 ↔ 〈𝑦, 𝑥〉 ∈ 𝐴) |
| 12 | 7, 8 | opelcnv 5892 | . . . . . 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 5800 | 1 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 〈cop 4632 ◡ccnv 5684 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: cnvin 6164 gtiso 32710 gsumhashmul 33064 mthmpps 35587 cnvnonrel 43601 |
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