Theorem elpwpw

Description: Characterization of the elements of a double power class: they are exactly the sets whose union is included in that class. (Contributed by BJ, 29-Apr-2021)

	Ref	Expression
Assertion	elpwpw	⊢ ( 𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ( 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		elpwb	⊢ ( 𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ( 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵 ) )
2		sspwuni	⊢ ( 𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵 )
3	2	anbi2i	⊢ ( ( 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵 ) ↔ ( 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵 ) )
4	1 3	bitri	⊢ ( 𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ( 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵 ) )