Theorem elpwpwel 7710

Description: A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.)

Assertion

Ref	Expression
elpwpwel	⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwpwel

Step	Hyp	Ref	Expression
1		uniexb 7707	. . 3 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
2	1	anbi1i 624	. 2 ⊢ ((𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵) ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
3		elpwpw 5055	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
4		elpwb 4560	. 2 ⊢ (∪ 𝐴 ∈ 𝒫 𝐵 ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
5	2, 3, 4	3bitr4i 303	1 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  Vcvv 3438   ⊆ wss 3899  𝒫 cpw 4552  ∪ cuni 4861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-pow 5308  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rab 3398  df-v 3440  df-in 3906  df-ss 3916  df-pw 4554  df-uni 4862

This theorem is referenced by:  elpwunicl  32578