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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-bj-moore | Structured version Visualization version GIF version |
Description: Define the class of Moore
collections. This is to df-mre 16246 what
df-top 20699 is to df-topon 20716. For the sake of consistency, the function
defined at df-mre 16246 should be denoted by "MooreOn".
Note: df-mre 16246 singles out the empty intersection. This is not necessary. It could be written instead Moore = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝒫 𝑥 ∣ ∀𝑧 ∈ 𝒫 𝑦(𝑥 ∩ ∩ 𝑧) ∈ 𝑦}) and the equivalence of both definitions is proved by bj-0int 33055. There is no added generality in defining a "Moore predicate" for arbitrary classes, since a Moore class satisfying such a predicate is automatically a set (see bj-mooreset 33056). (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
df-bj-moore | ⊢ Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmoore 33057 | . 2 class Moore | |
2 | vx | . . . . . . . 8 setvar 𝑥 | |
3 | 2 | cv 1482 | . . . . . . 7 class 𝑥 |
4 | 3 | cuni 4436 | . . . . . 6 class ∪ 𝑥 |
5 | vy | . . . . . . . 8 setvar 𝑦 | |
6 | 5 | cv 1482 | . . . . . . 7 class 𝑦 |
7 | 6 | cint 4475 | . . . . . 6 class ∩ 𝑦 |
8 | 4, 7 | cin 3573 | . . . . 5 class (∪ 𝑥 ∩ ∩ 𝑦) |
9 | 8, 3 | wcel 1990 | . . . 4 wff (∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥 |
10 | 3 | cpw 4158 | . . . 4 class 𝒫 𝑥 |
11 | 9, 5, 10 | wral 2912 | . . 3 wff ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥 |
12 | 11, 2 | cab 2608 | . 2 class {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥} |
13 | 1, 12 | wceq 1483 | 1 wff Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥} |
Colors of variables: wff setvar class |
This definition is referenced by: bj-ismoore 33059 |
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