Theorem bj-mooreset 37744

Description: A Moore collection is a set. Therefore, the class Moore of all Moore sets defined in df-bj-moore 37746 is actually the class of all Moore collections. This is also illustrated by the lack of sethood condition in bj-ismoore 37747.

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 7758).

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 5431), 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 37753. (Contributed by BJ, 8-Dec-2021.)

Assertion

Ref	Expression
bj-mooreset	⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V)

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-mooreset

Step	Hyp	Ref	Expression
1		0elpw 5326	. . 3 ⊢ ∅ ∈ 𝒫 𝐴
2		rint0 4953	. . . . 5 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴)
3	2	eleq1d 2848	. . . 4 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴))
4	3	rspcv 3577	. . 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 7758	. 2 ⊢ (∪ 𝐴 ∈ 𝐴 → 𝐴 ∈ V)
7	5, 6	syl 18	1 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∩ cin 3904  ∅c0 4286  𝒫 cpw 4562  ∪ cuni 4872  ∩ cint 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-in 3912  df-ss 3922  df-nul 4287  df-pw 4564  df-uni 4873  df-int 4913

This theorem is referenced by:  bj-ismoore  37747