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Theorem pwel 4836
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 4392 . . 3 (𝐴𝐵𝐴 𝐵)
2 sspwb 4833 . . 3 (𝐴 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
31, 2sylib 206 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
4 pwexg 4766 . . 3 (𝐴𝐵 → 𝒫 𝐴 ∈ V)
5 elpwg 4110 . . 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 245 1 (𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wcel 1975  Vcvv 3167  wss 3534  𝒫 cpw 4102   cuni 4361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-v 3169  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-pw 4104  df-sn 4120  df-pr 4122  df-uni 4362
This theorem is referenced by: (None)
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