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Theorem bj-mooreset 37103
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.)

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 5356 . . 3 ∅ ∈ 𝒫 𝐴
2 rint0 4988 . . . . 5 (𝑥 = ∅ → ( 𝐴 𝑥) = 𝐴)
32eleq1d 2826 . . . 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 7783 . 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 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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