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) ∈ 𝐴 → 𝐴 ∈ V)

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-mooreset

Step	Hyp	Ref	Expression
1		0elpw 5356	. . 3 ⊢ ∅ ∈ 𝒫 𝐴
2		rint0 4988	. . . . 5 ⊢ (𝑥 = ∅ → (∪ 𝐴 ∩ ∩ 𝑥) = ∪ 𝐴)
3	2	eleq1d 2826	. . . 4 ⊢ (𝑥 = ∅ → ((∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 ↔ ∪ 𝐴 ∈ 𝐴))
4	3	rspcv 3618	. . 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 7783	. 2 ⊢ (∪ 𝐴 ∈ 𝐴 → 𝐴 ∈ V)
7	5, 6	syl 17	1 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V)

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