Theorem bj-ismoore

Description: Characterization of Moore collections. Note that there is no sethood hypothesis on A : it is implied by either side (this is obvious for the LHS, and is the content of bj-mooreset for the RHS). (Contributed by BJ, 9-Dec-2021)

	Ref	Expression
Assertion	bj-ismoore	⊢ ( 𝐴 ∈ Moore ↔ ∀ 𝑥 ∈ 𝒫 𝐴 ( ∪ 𝐴 ∩ ∩ 𝑥 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ Moore → 𝐴 ∈ V )
2		bj-mooreset	⊢ ( ∀ 𝑥 ∈ 𝒫 𝐴 ( ∪ 𝐴 ∩ ∩ 𝑥 ) ∈ 𝐴 → 𝐴 ∈ V )
3		pweq	⊢ ( 𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴 )
4		unieq	⊢ ( 𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴 )
5	4	ineq1d	⊢ ( 𝑦 = 𝐴 → ( ∪ 𝑦 ∩ ∩ 𝑥 ) = ( ∪ 𝐴 ∩ ∩ 𝑥 ) )
6		id	⊢ ( 𝑦 = 𝐴 → 𝑦 = 𝐴 )
7	5 6	eleq12d	⊢ ( 𝑦 = 𝐴 → ( ( ∪ 𝑦 ∩ ∩ 𝑥 ) ∈ 𝑦 ↔ ( ∪ 𝐴 ∩ ∩ 𝑥 ) ∈ 𝐴 ) )
8	3 7	raleqbidv	⊢ ( 𝑦 = 𝐴 → ( ∀ 𝑥 ∈ 𝒫 𝑦 ( ∪ 𝑦 ∩ ∩ 𝑥 ) ∈ 𝑦 ↔ ∀ 𝑥 ∈ 𝒫 𝐴 ( ∪ 𝐴 ∩ ∩ 𝑥 ) ∈ 𝐴 ) )
9		df-bj-moore	⊢ Moore = { 𝑦 ∣ ∀ 𝑥 ∈ 𝒫 𝑦 ( ∪ 𝑦 ∩ ∩ 𝑥 ) ∈ 𝑦 }
10	8 9	elab2g	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ Moore ↔ ∀ 𝑥 ∈ 𝒫 𝐴 ( ∪ 𝐴 ∩ ∩ 𝑥 ) ∈ 𝐴 ) )
11	1 2 10	pm5.21nii	⊢ ( 𝐴 ∈ Moore ↔ ∀ 𝑥 ∈ 𝒫 𝐴 ( ∪ 𝐴 ∩ ∩ 𝑥 ) ∈ 𝐴 )