Theorem elpwpwel 4460

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 4458	. . 3 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
2	1	anbi1i 455	. 2 ⊢ ((𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵) ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
3		elpwpw 3959	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
4		elpwb 3576	. 2 ⊢ (∪ 𝐴 ∈ 𝒫 𝐵 ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
5	2, 3, 4	3bitr4i 211	1 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121  𝒫 cpw 3566  ∪ cuni 3796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-uni 3797

This theorem is referenced by: (None)