![]() |
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 6060 | . 2 ⊢ Rel ◡(𝐴 ∖ 𝐵) | |
2 | difss 4095 | . . 3 ⊢ (◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 | |
3 | relcnv 6060 | . . 3 ⊢ Rel ◡𝐴 | |
4 | relss 5741 | . . 3 ⊢ ((◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 → (Rel ◡𝐴 → Rel (◡𝐴 ∖ ◡𝐵))) | |
5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel (◡𝐴 ∖ ◡𝐵) |
6 | eldif 3924 | . . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)) | |
7 | vex 3451 | . . . 4 ⊢ 𝑥 ∈ V | |
8 | vex 3451 | . . . 4 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | opelcnv 5841 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∖ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵)) |
10 | eldif 3924 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵)) | |
11 | 7, 8 | opelcnv 5841 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴) |
12 | 7, 8 | opelcnv 5841 | . . . . . 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 5750 | 1 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3911 ⊆ wss 3914 ⟨cop 4596 ◡ccnv 5636 Rel wrel 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 |
This theorem is referenced by: cnvin 6101 gtiso 31668 gsumhashmul 31954 mthmpps 34240 cnvnonrel 41952 |
Copyright terms: Public domain | W3C validator |