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Theorem pwel 5387
Description: Quantitative version of pwexg 5384: 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 5312 and ax-pr 5438 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 5384 . 2 (𝐴𝐵 → 𝒫 𝐴 ∈ V)
2 elssuni 4942 . . 3 (𝐴𝐵𝐴 𝐵)
32sspwd 4618 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
41, 3elpwd 4611 1 (𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3478  𝒫 cpw 4605   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pow 5371
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-pw 4607  df-uni 4913
This theorem is referenced by:  bj-unirel  37034
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