Theorem bj-ismoorec 33060

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

Assertion

Ref	Expression
bj-ismoorec	|- ( A e. Moore_ <-> ( A e. _V /\ A. x e. ~P A ( U. A i^i |^| x ) e. A ) )

Distinct variable group:   x, A

Proof of Theorem bj-ismoorec

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. Moore_ -> A e. _V )
2		bj-ismoore 33059	. 2 |- ( A e. _V -> ( A e. Moore_ <-> A. x e. ~P A ( U. A i^i |^| x ) e. A ) )
3	1, 2	biadan2 674	1 |- ( A e. Moore_ <-> ( A e. _V /\ A. x e. ~P A ( U. A i^i |^| x ) e. A ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   _Vcvv 3200   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-ismoored  33062