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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 36636 is actually the class of all Moore
collections. This is also illustrated by the lack of sethood condition
in bj-ismoore 36637.
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 7760). 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 5447), 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 36643. (Contributed by BJ, 8-Dec-2021.) |
Ref | Expression |
---|---|
bj-mooreset | ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 5351 | . . 3 ⊢ ∅ ∈ 𝒫 𝐴 | |
2 | rint0 4989 | . . . . 5 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴) | |
3 | 2 | eleq1d 2810 | . . . 4 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴)) |
4 | 3 | rspcv 3599 | . . 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 7760 | . 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 1533 ∈ wcel 2098 ∀wral 3051 Vcvv 3463 ∩ cin 3940 ∅c0 4319 𝒫 cpw 4599 ∪ cuni 4904 ∩ cint 4945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3944 df-in 3948 df-ss 3958 df-nul 4320 df-pw 4601 df-uni 4905 df-int 4946 |
This theorem is referenced by: bj-ismoore 36637 |
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