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Theorem bj-sspwpwab 32692
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 1840 . . 3 𝑥
2 bj-nfab1 32425 . . 3 𝑥{𝑥 𝑥𝐴}
3 nfcv 2761 . . 3 𝑥𝒫 𝒫 𝐴
4 abid 2609 . . . . 5 (𝑥 ∈ {𝑥 𝑥𝐴} ↔ 𝑥𝐴)
5 vex 3189 . . . . . 6 𝑥 ∈ V
65biantrur 527 . . . . 5 ( 𝑥𝐴 ↔ (𝑥 ∈ V ∧ 𝑥𝐴))
7 bj-sspwpw 32691 . . . . 5 ((𝑥 ∈ V ∧ 𝑥𝐴) ↔ 𝑥 ∈ 𝒫 𝒫 𝐴)
84, 6, 73bitri 286 . . . 4 (𝑥 ∈ {𝑥 𝑥𝐴} ↔ 𝑥 ∈ 𝒫 𝒫 𝐴)
98a1i 11 . . 3 (⊤ → (𝑥 ∈ {𝑥 𝑥𝐴} ↔ 𝑥 ∈ 𝒫 𝒫 𝐴))
101, 2, 3, 9eqrd 3602 . 2 (⊤ → {𝑥 𝑥𝐴} = 𝒫 𝒫 𝐴)
1110trud 1490 1 {𝑥 𝑥𝐴} = 𝒫 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wtru 1481  wcel 1987  {cab 2607  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-in 3562  df-ss 3569  df-pw 4132  df-uni 4403
This theorem is referenced by:  bj-sspwpweq  32693
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