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Theorem pwel 5198
 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 5129 . 2 (𝐴𝐵 → 𝒫 𝐴 ∈ V)
2 elssuni 4738 . . 3 (𝐴𝐵𝐴 𝐵)
3 sspwb 5195 . . 3 (𝐴 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
42, 3sylib 210 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
51, 4elpwd 4426 1 (𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2051  Vcvv 3410   ⊆ wss 3824  𝒫 cpw 4417  ∪ cuni 4709 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-pw 4419  df-sn 4437  df-pr 4439  df-uni 4710 This theorem is referenced by: (None)
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