Theorem bj-ismoored0 33061

Description: Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)

Assertion

Ref	Expression
bj-ismoored0	|- ( A e. Moore_ -> U. A e. A )

Proof of Theorem bj-ismoored0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-ismoore 33059	. . 3 |- ( A e. Moore_ -> ( A e. Moore_ <-> A. x e. ~P A ( U. A i^i |^| x ) e. A ) )
2		0elpw 4834	. . . 4 |- (/) e. ~P A
3		rint0 4517	. . . . . 6 |- ( x = (/) -> ( U. A i^i |^| x ) = U. A )
4	3	eleq1d 2686	. . . . 5 |- ( x = (/) -> ( ( U. A i^i |^| x ) e. A <-> U. A e. A ) )
5	4	rspcv 3305	. . . 4 |- ( (/) e. ~P A -> ( A. x e. ~P A ( U. A i^i |^| x ) e. A -> U. A e. A ) )
6	2, 5	ax-mp 5	. . 3 |- ( A. x e. ~P A ( U. A i^i |^| x ) e. A -> U. A e. A )
7	1, 6	syl6bi 243	. 2 |- ( A e. Moore_ -> ( A e. Moore_ -> U. A e. A ) )
8	7	pm2.43i 52	1 |- ( A e. Moore_ -> U. A e. A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   (/)c0 3915   ~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  ax-nul 4789

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-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-uni 4437  df-int 4476  df-bj-moore 33058

This theorem is referenced by:  bj-0nmoore  33067