Theorem pwpwab 5107

Description: The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
pwpwab	⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}

Distinct variable group:   𝑥,𝐴

Proof of Theorem pwpwab

Step	Hyp	Ref	Expression
1		vex 3465	. . 3 ⊢ 𝑥 ∈ V
2		elpwpw 5106	. . 3 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ ∪ 𝑥 ⊆ 𝐴))
3	1, 2	mpbiran 707	. 2 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
4	3	eqabi 2861	1 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}

Syntax hints:   = wceq 1533   ∈ wcel 2098  {cab 2702  Vcvv 3461   ⊆ wss 3944  𝒫 cpw 4604  ∪ cuni 4909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-v 3463  df-ss 3961  df-pw 4606  df-uni 4910

This theorem is referenced by:  pwpwssunieq  5108