Theorem bj-ismoore 33059

Description: Characterization of Moore collections among sets. (Contributed by BJ, 9-Dec-2021.)

Assertion

Ref	Expression
bj-ismoore	|- ( A e. V -> ( A e. Moore_ <-> A. x e. ~P A ( U. A i^i |^| x ) e. A ) )

Distinct variable group:   x, A

Allowed substitution hint:   V( x)

Proof of Theorem bj-ismoore

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4161	. . 3 |- ( y = A -> ~P y = ~P A )
2		unieq 4444	. . . . 5 |- ( y = A -> U. y = U. A )
3	2	ineq1d 3813	. . . 4 |- ( y = A -> ( U. y i^i |^| x ) = ( U. A i^i |^| x ) )
4		id 22	. . . 4 |- ( y = A -> y = A )
5	3, 4	eleq12d 2695	. . 3 |- ( y = A -> ( ( U. y i^i |^| x ) e. y <-> ( U. A i^i |^| x ) e. A ) )
6	1, 5	raleqbidv 3152	. 2 |- ( y = A -> ( A. x e. ~P y ( U. y i^i |^| x ) e. y <-> A. x e. ~P A ( U. A i^i |^| x ) e. A ) )
7		df-bj-moore 33058	. 2 |- Moore_ = { y | A. x e. ~P y ( U. y i^i |^| x ) e. y }
8	6, 7	elab2g 3353	1 |- ( A e. V -> ( A e. Moore_ <-> A. x e. ~P A ( U. A i^i |^| x ) e. A ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   ~Pcpw 4158   U.cuni 4436   |^|cint 4475  Moore_cmoore 33057

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

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-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-bj-moore 33058

This theorem is referenced by:  bj-ismoorec  33060  bj-ismoored0  33061  bj-ismooredr  33064  bj-ismooredr2  33065