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Theorem pwpwab 5103
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 3484 . . 3 𝑥 ∈ V
2 elpwpw 5102 . . 3 (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ 𝑥𝐴))
31, 2mpbiran 709 . 2 (𝑥 ∈ 𝒫 𝒫 𝐴 𝑥𝐴)
43eqabi 2877 1 𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  wss 3951  𝒫 cpw 4600   cuni 4907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-ss 3968  df-pw 4602  df-uni 4908
This theorem is referenced by:  pwpwssunieq  5104
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