Theorem bj-sspwpw 32238

Description: The union of a set is included in a given class if and only if that set is an element of the double power class of that class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
bj-sspwpw	⊢ ((𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵) ↔ 𝐴 ∈ 𝒫 𝒫 𝐵)

Proof of Theorem bj-sspwpw

Step	Hyp	Ref	Expression
1		sspwuni 4547	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
2	1	bicomi 213	. . 3 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵)
3	2	anbi2i 726	. 2 ⊢ ((𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵) ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵))
4		bj-elpw3 32237	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵) ↔ 𝐴 ∈ 𝒫 𝒫 𝐵)
5	3, 4	bitri 263	1 ⊢ ((𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵) ↔ 𝐴 ∈ 𝒫 𝒫 𝐵)

Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373

This theorem is referenced by:  bj-sspwpwab  32239