Theorem elpwpwel 4578

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 4576	. . 3 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
2	1	anbi1i 458	. 2 ⊢ ((𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵) ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
3		elpwpw 4062	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
4		elpwb 3666	. 2 ⊢ (∪ 𝐴 ∈ 𝒫 𝐵 ↔ (∪ 𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
5	2, 3, 4	3bitr4i 212	1 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2202  Vcvv 2803   ⊆ wss 3201  𝒫 cpw 3656  ∪ cuni 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658  df-uni 3899

This theorem is referenced by: (None)