Theorem bj-ismoored 33062

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

Hypotheses

Ref	Expression
bj-ismoored.1	|- ( ph -> A e. Moore_ )
bj-ismoored.2	|- ( ph -> B C_ A )

Assertion

Ref	Expression
bj-ismoored	|- ( ph -> ( U. A i^i |^| B ) e. A )

Proof of Theorem bj-ismoored

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-ismoored.2	. 2 |- ( ph -> B C_ A )
2		bj-ismoored.1	. . 3 |- ( ph -> A e. Moore_ )
3		bj-ismoorec 33060	. . 3 |- ( A e. Moore_ <-> ( A e. _V /\ A. x e. ~P A ( U. A i^i |^| x ) e. A ) )
4	2, 3	sylib 208	. 2 |- ( ph -> ( A e. _V /\ A. x e. ~P A ( U. A i^i |^| x ) e. A ) )
5		elpw2g 4827	. . . . 5 |- ( A e. _V -> ( B e. ~P A <-> B C_ A ) )
6	5	biimparc 504	. . . 4 |- ( ( B C_ A /\ A e. _V ) -> B e. ~P A )
7		inteq 4478	. . . . . . 7 |- ( x = B -> |^| x = |^| B )
8	7	ineq2d 3814	. . . . . 6 |- ( x = B -> ( U. A i^i |^| x ) = ( U. A i^i |^| B ) )
9	8	eleq1d 2686	. . . . 5 |- ( x = B -> ( ( U. A i^i |^| x ) e. A <-> ( U. A i^i |^| B ) e. A ) )
10	9	rspcv 3305	. . . 4 |- ( B e. ~P A -> ( A. x e. ~P A ( U. A i^i |^| x ) e. A -> ( U. A i^i |^| B ) e. A ) )
11	6, 10	syl 17	. . 3 |- ( ( B C_ A /\ A e. _V ) -> ( A. x e. ~P A ( U. A i^i |^| x ) e. A -> ( U. A i^i |^| B ) e. A ) )
12	11	expimpd 629	. 2 |- ( B C_ A -> ( ( A e. _V /\ A. x e. ~P A ( U. A i^i |^| x ) e. A ) -> ( U. A i^i |^| B ) e. A ) )
13	1, 4, 12	sylc 65	1 |- ( ph -> ( U. A i^i |^| B ) e. A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~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-sep 4781

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-int 4476  df-bj-moore 33058

This theorem is referenced by:  bj-ismoored2  33063