Theorem pwpwab 5069

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 3459	. . 3 ⊢ 𝑥 ∈ V
2		elpwpw 5068	. . 3 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ ∪ 𝑥 ⊆ 𝐴))
3	1, 2	mpbiran 721	. 2 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
4	3	eqabi 2898	1 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}

Syntax hints:   = wceq 1570   ∈ wcel 2143  {cab 2741  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562  ∪ cuni 4872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-ss 3922  df-pw 4564  df-uni 4873

This theorem is referenced by:  pwpwssunieq  5070