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Theorem pwexb 7768
Description: The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
pwexb (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Proof of Theorem pwexb
StepHypRef Expression
1 pwexg 5378 . 2 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2 pwexr 7767 . 2 (𝒫 𝐴 ∈ V → 𝐴 ∈ V)
31, 2impbii 208 1 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2099  Vcvv 3471  𝒫 cpw 4603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-un 3952  df-in 3954  df-ss 3964  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4909
This theorem is referenced by:  2pwuninel  9156  ranklim  9867  r1pwALT  9869  isf34lem6  10403  isfin1-2  10408  pwfseqlem4  10685  pwfseqlem5  10686  gchpwdom  10693  hargch  10696  numufl  23818
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