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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-mooreset | Structured version Visualization version GIF version | ||
| Description: A Moore collection is a
set.  Therefore, the class Moore of all
       Moore sets defined in df-bj-moore 37105 is actually the class of all Moore
       collections.  This is also illustrated by the lack of sethood condition
       in bj-ismoore 37106. Note that the closed sets of a topology form a Moore collection, so a topology is a set, and this remark also applies to many other families of sets (namely, as soon as the whole set is required to be a set of the family, then the associated kind of family has no proper classes: that this condition suffices to impose sethood can be seen in this proof, which relies crucially on uniexr 7783). Note: if, in the above predicate, we substitute 𝒫 𝑋 for 𝐴, then the last ∈ 𝒫 𝑋 could be weakened to ⊆ 𝑋, and then the predicate would be obviously satisfied since ⊢ ∪ 𝒫 𝑋 = 𝑋 (unipw 5455), making 𝒫 𝑋 a Moore collection in this weaker sense, for any class 𝑋, even proper, but the addition of this single case does not add anything interesting. Instead, we have the biconditional bj-discrmoore 37112. (Contributed by BJ, 8-Dec-2021.) | 
| Ref | Expression | 
|---|---|
| bj-mooreset | ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0elpw 5356 | . . 3 ⊢ ∅ ∈ 𝒫 𝐴 | |
| 2 | rint0 4988 | . . . . 5 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴) | |
| 3 | 2 | eleq1d 2826 | . . . 4 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴)) | 
| 4 | 3 | rspcv 3618 | . . 3 ⊢ (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → ∪ 𝐴 ∈ 𝐴)) | 
| 5 | 1, 4 | ax-mp 5 | . 2 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → ∪ 𝐴 ∈ 𝐴) | 
| 6 | uniexr 7783 | . 2 ⊢ (∪ 𝐴 ∈ 𝐴 → 𝐴 ∈ V) | |
| 7 | 5, 6 | syl 17 | 1 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∩ cin 3950 ∅c0 4333 𝒫 cpw 4600 ∪ cuni 4907 ∩ cint 4946 | 
| 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 | 
| This theorem is referenced by: bj-ismoore 37106 | 
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