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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ismooredr | Structured version Visualization version GIF version | ||
| Description: Sufficient condition to be a Moore collection. Note that there is no sethood hypothesis on 𝐴: it is a consequence of the only hypothesis. (Contributed by BJ, 9-Dec-2021.) | 
| Ref | Expression | 
|---|---|
| bj-ismooredr.1 | ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) | 
| Ref | Expression | 
|---|---|
| bj-ismooredr | ⊢ (𝜑 → 𝐴 ∈ Moore) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elpwi 4607 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
| 2 | bj-ismooredr.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) | |
| 3 | 2 | ex 412 | . . . 4 ⊢ (𝜑 → (𝑥 ⊆ 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) | 
| 4 | 1, 3 | syl5 34 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) | 
| 5 | 4 | ralrimiv 3145 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) | 
| 6 | bj-ismoore 37106 | . 2 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) | |
| 7 | 5, 6 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 ∈ Moore) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 ⊆ wss 3951 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-ismooredr2 37111 bj-discrmoore 37112 | 
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