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Theorem pwpwab 5108
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 𝑥𝐴}
Distinct variable group:   𝑥,𝐴

Proof of Theorem pwpwab
StepHypRef Expression
1 vex 3482 . . 3 𝑥 ∈ V
2 elpwpw 5107 . . 3 (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ 𝑥𝐴))
31, 2mpbiran 709 . 2 (𝑥 ∈ 𝒫 𝒫 𝐴 𝑥𝐴)
43eqabi 2875 1 𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478  wss 3963  𝒫 cpw 4605   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-ss 3980  df-pw 4607  df-uni 4913
This theorem is referenced by:  pwpwssunieq  5109
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