Theorem bj-mooreset 33056

Description: A Moore collection is a set. That is, if we define a "Moore predicate" by (Moore A <-> A. x e. ~P A ( U. A i^i |^| x ) e. A ), then any class satisfying that predicate is actually a set. Therefore, the definition df-bj-moore 33058 is sufficient. Note that the closed sets of a topology form a Moore collection, so this remark also applies to topologies and many other families of sets (namely, as soon as the whole set is required to be a closed set, as can be seen from the proof, which relies crucially on uniexr 6972).

Note: if, in the above predicate, we substitute ~P X for A, then the last e. ~P X could be weakened to C_ ~P X, and then the predicate would be obviously satisfied since U. ~P X = X, making ~P X a Moore collection in this weaker sense, even if X is a proper class, but the addition of this single case does not add anything interesting. (Contributed by BJ, 8-Dec-2021.)

Assertion

Ref	Expression
bj-mooreset	|- ( A. x e. ~P A ( U. A i^i |^| x ) e. A -> A e. _V )

Distinct variable group:   x, A

Proof of Theorem bj-mooreset

Step	Hyp	Ref	Expression
1		0elpw 4834	. . 3 |- (/) e. ~P A
2		rint0 4517	. . . . 5 |- ( x = (/) -> ( U. A i^i |^| x ) = U. A )
3	2	eleq1d 2686	. . . 4 |- ( x = (/) -> ( ( U. A i^i |^| x ) e. A <-> U. A e. A ) )
4	3	rspcv 3305	. . 3 |- ( (/) e. ~P A -> ( A. x e. ~P A ( U. A i^i |^| x ) e. A -> U. A e. A ) )
5	1, 4	ax-mp 5	. 2 |- ( A. x e. ~P A ( U. A i^i |^| x ) e. A -> U. A e. A )
6		uniexr 6972	. 2 |- ( U. A e. A -> A e. _V )
7	5, 6	syl 17	1 |- ( A. x e. ~P A ( U. A i^i |^| x ) e. A -> A e. _V )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-uni 4437  df-int 4476

This theorem is referenced by: (None)