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Theorem pwel 5110
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 elssuni 4661 . . 3 (𝐴𝐵𝐴 𝐵)
2 sspwb 5107 . . 3 (𝐴 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
31, 2sylib 209 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
4 pwexg 5048 . . 3 (𝐴𝐵 → 𝒫 𝐴 ∈ V)
5 elpwg 4359 . . 3 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵))
64, 5syl 17 . 2 (𝐴𝐵 → (𝒫 𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵))
73, 6mpbird 248 1 (𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wcel 2156  Vcvv 3391  wss 3769  𝒫 cpw 4351   cuni 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-pw 4353  df-sn 4371  df-pr 4373  df-uni 4631
This theorem is referenced by: (None)
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