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Theorem bj-mooreset 37085
Description: A Moore collection is a set. Therefore, the class Moore of all Moore sets defined in df-bj-moore 37087 is actually the class of all Moore collections. This is also illustrated by the lack of sethood condition in bj-ismoore 37088.

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 7782).

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 5461), 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 37094. (Contributed by BJ, 8-Dec-2021.)

Assertion
Ref Expression
bj-mooreset (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴𝐴 ∈ V)
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-mooreset
StepHypRef Expression
1 0elpw 5362 . . 3 ∅ ∈ 𝒫 𝐴
2 rint0 4993 . . . . 5 (𝑥 = ∅ → ( 𝐴 𝑥) = 𝐴)
32eleq1d 2824 . . . 4 (𝑥 = ∅ → (( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
43rspcv 3618 . . 3 (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
51, 4ax-mp 5 . 2 (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴)
6 uniexr 7782 . 2 ( 𝐴𝐴𝐴 ∈ V)
75, 6syl 17 1 (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cin 3962  c0 4339  𝒫 cpw 4605   cuni 4912   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-uni 4913  df-int 4952
This theorem is referenced by:  bj-ismoore  37088
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