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Theorem bj-sspwpwab 32239
 Description: The class of families whose union is included in a given class is equal to the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
Assertion
Ref Expression
bj-sspwpwab {𝑥 𝑥𝐴} = 𝒫 𝒫 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-sspwpwab
StepHypRef Expression
1 nfv 1830 . . 3 𝑥
2 bj-nfab1 31973 . . 3 𝑥{𝑥 𝑥𝐴}
3 nfcv 2751 . . 3 𝑥𝒫 𝒫 𝐴
4 abid 2598 . . . . 5 (𝑥 ∈ {𝑥 𝑥𝐴} ↔ 𝑥𝐴)
5 vex 3176 . . . . . 6 𝑥 ∈ V
65biantrur 526 . . . . 5 ( 𝑥𝐴 ↔ (𝑥 ∈ V ∧ 𝑥𝐴))
7 bj-sspwpw 32238 . . . . 5 ((𝑥 ∈ V ∧ 𝑥𝐴) ↔ 𝑥 ∈ 𝒫 𝒫 𝐴)
84, 6, 73bitri 285 . . . 4 (𝑥 ∈ {𝑥 𝑥𝐴} ↔ 𝑥 ∈ 𝒫 𝒫 𝐴)
98a1i 11 . . 3 (⊤ → (𝑥 ∈ {𝑥 𝑥𝐴} ↔ 𝑥 ∈ 𝒫 𝒫 𝐴))
101, 2, 3, 9eqrd 3586 . 2 (⊤ → {𝑥 𝑥𝐴} = 𝒫 𝒫 𝐴)
1110trud 1484 1 {𝑥 𝑥𝐴} = 𝒫 𝒫 𝐴
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  {cab 2596  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373 This theorem is referenced by:  bj-sspwpweq  32240
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