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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ismoore | Structured version Visualization version GIF version |
Description: Characterization of Moore collections. Note that there is no sethood hypothesis on 𝐴: it is implied by either side (this is obvious for the LHS, and is the content of bj-mooreset 35386 for the RHS). (Contributed by BJ, 9-Dec-2021.) |
Ref | Expression |
---|---|
bj-ismoore | ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3459 | . 2 ⊢ (𝐴 ∈ Moore → 𝐴 ∈ V) | |
2 | bj-mooreset 35386 | . 2 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V) | |
3 | pweq 4561 | . . . 4 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴) | |
4 | unieq 4863 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴) | |
5 | 4 | ineq1d 4158 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∪ 𝑦 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝑥)) |
6 | id 22 | . . . . 5 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴) | |
7 | 5, 6 | eleq12d 2831 | . . . 4 ⊢ (𝑦 = 𝐴 → ((∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) |
8 | 3, 7 | raleqbidv 3315 | . . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) |
9 | df-bj-moore 35388 | . . 3 ⊢ Moore = {𝑦 ∣ ∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦} | |
10 | 8, 9 | elab2g 3621 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) |
11 | 1, 2, 10 | pm5.21nii 379 | 1 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1540 ∈ wcel 2105 ∀wral 3061 Vcvv 3441 ∩ cin 3897 𝒫 cpw 4547 ∪ cuni 4852 ∩ cint 4894 Moorecmoore 35387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3062 df-rab 3404 df-v 3443 df-dif 3901 df-in 3905 df-ss 3915 df-nul 4270 df-pw 4549 df-uni 4853 df-int 4895 df-bj-moore 35388 |
This theorem is referenced by: bj-ismoored0 35390 bj-ismoored 35391 bj-ismooredr 35393 |
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