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Theorem pwpwab 4805
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 3388 . . 3 𝑥 ∈ V
2 elpwpw 4804 . . 3 (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ 𝑥𝐴))
31, 2mpbiran 701 . 2 (𝑥 ∈ 𝒫 𝒫 𝐴 𝑥𝐴)
43abbi2i 2915 1 𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wcel 2157  {cab 2785  Vcvv 3385  wss 3769  𝒫 cpw 4349   cuni 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387  df-in 3776  df-ss 3783  df-pw 4351  df-uni 4629
This theorem is referenced by:  pwpwssunieq  4806
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