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Theorem pwel 5375
Description: Quantitative version of pwexg 5372: the powerset of an element of a class is an element of the double powerclass of the union of that class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) Remove use of ax-nul 5300 and ax-pr 5423 and shorten proof. (Revised by BJ, 13-Apr-2024.)
Assertion
Ref Expression
pwel (𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)

Proof of Theorem pwel
StepHypRef Expression
1 pwexg 5372 . 2 (𝐴𝐵 → 𝒫 𝐴 ∈ V)
2 elssuni 4935 . . 3 (𝐴𝐵𝐴 𝐵)
32sspwd 4611 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
41, 3elpwd 4604 1 (𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  Vcvv 3469  𝒫 cpw 4598   cuni 4903
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 2698  ax-sep 5293  ax-pow 5359
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-in 3951  df-ss 3961  df-pw 4600  df-uni 4904
This theorem is referenced by:  bj-unirel  36466
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