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Theorem pwexb 7785
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 5384 . 2 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2 pwexr 7784 . 2 (𝒫 𝐴 ∈ V → 𝐴 ∈ V)
31, 2impbii 209 1 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2106  Vcvv 3478  𝒫 cpw 4605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913
This theorem is referenced by:  2pwuninel  9171  ranklim  9882  r1pwALT  9884  isf34lem6  10418  isfin1-2  10423  pwfseqlem4  10700  pwfseqlem5  10701  gchpwdom  10708  hargch  10711  numufl  23939
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