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Theorem pwpwab 5106
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 3475 . . 3 𝑥 ∈ V
2 elpwpw 5105 . . 3 (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ 𝑥𝐴))
31, 2mpbiran 708 . 2 (𝑥 ∈ 𝒫 𝒫 𝐴 𝑥𝐴)
43eqabi 2865 1 𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  {cab 2705  Vcvv 3471  wss 3947  𝒫 cpw 4603   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-v 3473  df-in 3954  df-ss 3964  df-pw 4605  df-uni 4909
This theorem is referenced by:  pwpwssunieq  5107
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