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Theorem pwel 5335
Description: Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
Assertion
Ref Expression
pwel (𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)

Proof of Theorem pwel
StepHypRef Expression
1 pwexg 5270 . 2 (𝐴𝐵 → 𝒫 𝐴 ∈ V)
2 elssuni 4859 . . 3 (𝐴𝐵𝐴 𝐵)
3 sspwb 5332 . . 3 (𝐴 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
42, 3sylib 219 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
51, 4elpwd 4546 1 (𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  Vcvv 3492  wss 3933  𝒫 cpw 4535   cuni 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-pw 4537  df-sn 4558  df-pr 4560  df-uni 4831
This theorem is referenced by: (None)
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