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Theorem pwpwab 5107
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 3465 . . 3 𝑥 ∈ V
2 elpwpw 5106 . . 3 (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ 𝑥𝐴))
31, 2mpbiran 707 . 2 (𝑥 ∈ 𝒫 𝒫 𝐴 𝑥𝐴)
43eqabi 2861 1 𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  {cab 2702  Vcvv 3461  wss 3944  𝒫 cpw 4604   cuni 4909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-v 3463  df-ss 3961  df-pw 4606  df-uni 4910
This theorem is referenced by:  pwpwssunieq  5108
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