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Theorem pwexb 7747
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 5367 . 2 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2 pwexr 7746 . 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 2098  Vcvv 3466  𝒫 cpw 4595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3946  df-in 3948  df-ss 3958  df-pw 4597  df-sn 4622  df-pr 4624  df-uni 4901
This theorem is referenced by:  2pwuninel  9129  ranklim  9836  r1pwALT  9838  isf34lem6  10372  isfin1-2  10377  pwfseqlem4  10654  pwfseqlem5  10655  gchpwdom  10662  hargch  10665  numufl  23763
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