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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 37103 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 3501 | . 2 ⊢ (𝐴 ∈ Moore → 𝐴 ∈ V) | |
| 2 | bj-mooreset 37103 | . 2 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V) | |
| 3 | pweq 4614 | . . . 4 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴) | |
| 4 | unieq 4918 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴) | |
| 5 | 4 | ineq1d 4219 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∪ 𝑦 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝑥)) |
| 6 | id 22 | . . . . 5 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴) | |
| 7 | 5, 6 | eleq12d 2835 | . . . 4 ⊢ (𝑦 = 𝐴 → ((∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) |
| 8 | 3, 7 | raleqbidv 3346 | . . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) |
| 9 | df-bj-moore 37105 | . . 3 ⊢ Moore = {𝑦 ∣ ∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦} | |
| 10 | 8, 9 | elab2g 3680 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) |
| 11 | 1, 2, 10 | pm5.21nii 378 | 1 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∩ cin 3950 𝒫 cpw 4600 ∪ cuni 4907 ∩ cint 4946 Moorecmoore 37104 |
| 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-pow 5365 |
| This theorem depends on definitions: df-bi 207 df-an 396 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-uni 4908 df-int 4947 df-bj-moore 37105 |
| This theorem is referenced by: bj-ismoored0 37107 bj-ismoored 37108 bj-ismooredr 37110 |
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