Theorem elpwpwel 7766

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 7763	. . 3 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
2	1	anbi1i 635	. 2 ⊢ ((𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵) ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
3		elpwpw 5072	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
4		elpwb 4575	. 2 ⊢ (∪ 𝐴 ∈ 𝒫 𝐵 ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
5	2, 3, 4	3bitr4i 306	1 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913  𝒫 cpw 4567  ∪ cuni 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pow 5337  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-in 3920  df-ss 3930  df-pw 4569  df-uni 4877

This theorem is referenced by:  elpwunicl  32840