Theorem bj-sspwpwab 32239

Description: The class of families whose union is included in a given class is equal to the double power class of that class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
bj-sspwpwab	⊢ {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} = 𝒫 𝒫 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-sspwpwab

Step	Hyp	Ref	Expression
1		nfv 1830	. . 3 ⊢ Ⅎ𝑥⊤
2		bj-nfab1 31973	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}
3		nfcv 2751	. . 3 ⊢ Ⅎ𝑥𝒫 𝒫 𝐴
4		abid 2598	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} ↔ ∪ 𝑥 ⊆ 𝐴)
5		vex 3176	. . . . . 6 ⊢ 𝑥 ∈ V
6	5	biantrur 526	. . . . 5 ⊢ (∪ 𝑥 ⊆ 𝐴 ↔ (𝑥 ∈ V ∧ ∪ 𝑥 ⊆ 𝐴))
7		bj-sspwpw 32238	. . . . 5 ⊢ ((𝑥 ∈ V ∧ ∪ 𝑥 ⊆ 𝐴) ↔ 𝑥 ∈ 𝒫 𝒫 𝐴)
8	4, 6, 7	3bitri 285	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} ↔ 𝑥 ∈ 𝒫 𝒫 𝐴)
9	8	a1i 11	. . 3 ⊢ (⊤ → (𝑥 ∈ {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} ↔ 𝑥 ∈ 𝒫 𝒫 𝐴))
10	1, 2, 3, 9	eqrd 3586	. 2 ⊢ (⊤ → {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} = 𝒫 𝒫 𝐴)
11	10	trud 1484	1 ⊢ {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} = 𝒫 𝒫 𝐴

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  {cab 2596  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-sspwpweq  32240