Theorem bj-ismooredr 33064

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

Hypotheses

Ref	Expression
bj-ismooredr.1	|- ( ph -> A e. V )
bj-ismooredr.2	|- ( ( ph /\ x C_ A ) -> ( U. A i^i |^| x ) e. A )

Assertion

Ref	Expression
bj-ismooredr	|- ( ph -> A e. Moore_ )

Distinct variable groups:   ph, x   x, A

Allowed substitution hint:   V( x)

Proof of Theorem bj-ismooredr

Step	Hyp	Ref	Expression
1		elpwi 4168	. . . . 5 |- ( x e. ~P A -> x C_ A )
2		bj-ismooredr.2	. . . . . 6 |- ( ( ph /\ x C_ A ) -> ( U. A i^i |^| x ) e. A )
3	2	ex 450	. . . . 5 |- ( ph -> ( x C_ A -> ( U. A i^i |^| x ) e. A ) )
4	1, 3	syl5 34	. . . 4 |- ( ph -> ( x e. ~P A -> ( U. A i^i |^| x ) e. A ) )
5	4	alrimiv 1855	. . 3 |- ( ph -> A. x ( x e. ~P A -> ( U. A i^i |^| x ) e. A ) )
6		df-ral 2917	. . 3 |- ( A. x e. ~P A ( U. A i^i |^| x ) e. A <-> A. x ( x e. ~P A -> ( U. A i^i |^| x ) e. A ) )
7	5, 6	sylibr 224	. 2 |- ( ph -> A. x e. ~P A ( U. A i^i |^| x ) e. A )
8		bj-ismooredr.1	. . 3 |- ( ph -> A e. V )
9		bj-ismoore 33059	. . 3 |- ( A e. V -> ( A e. Moore_ <-> A. x e. ~P A ( U. A i^i |^| x ) e. A ) )
10	8, 9	syl 17	. 2 |- ( ph -> ( A e. Moore_ <-> A. x e. ~P A ( U. A i^i |^| x ) e. A ) )
11	7, 10	mpbird 247	1 |- ( ph -> A e. Moore_ )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   A.wral 2912   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

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-discrmoore  33066