Theorem bj-ismooredr

Description: Sufficient condition to be a Moore collection. Note that there is no sethood hypothesis on A : it is a consequence of the only hypothesis. (Contributed by BJ, 9-Dec-2021)

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

Proof

Step	Hyp	Ref	Expression
1		bj-ismooredr.1	|- ( ( ph /\ x C_ A ) -> ( U. A i^i |^| x ) e. A )
2		elpwi	|- ( x e. ~P A -> x C_ A )
3	1	ex	|- ( ph -> ( x C_ A -> ( U. A i^i |^| x ) e. A ) )
4	2 3	syl5	|- ( ph -> ( x e. ~P A -> ( U. A i^i |^| x ) e. A ) )
5	4	ralrimiv	|- ( ph -> A. x e. ~P A ( U. A i^i |^| x ) e. A )
6		bj-ismoore	|- ( A e. Moore_ <-> A. x e. ~P A ( U. A i^i |^| x ) e. A )
7	5 6	sylibr	|- ( ph -> A e. Moore_ )