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Theorem bj-mooreset 34536
 Description: A Moore collection is a set. Therefore, the class Moore of all Moore sets defined in df-bj-moore 34538 is actually the class of all Moore collections. This is also illustrated by the lack of sethood condition in bj-ismoore 34539. 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 7468). 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 5309) , 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 34545. (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 5222 . . 3 ∅ ∈ 𝒫 𝐴
2 rint0 4879 . . . . 5 (𝑥 = ∅ → ( 𝐴 𝑥) = 𝐴)
32eleq1d 2874 . . . 4 (𝑥 = ∅ → (( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
43rspcv 3566 . . 3 (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
51, 4ax-mp 5 . 2 (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴)
6 uniexr 7468 . 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 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∩ cin 3880  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4801  ∩ cint 4839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244  df-pw 4499  df-uni 4802  df-int 4840 This theorem is referenced by:  bj-ismoore  34539
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