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Theorem pwexb 7471
 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 5260 . 2 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2 pwexr 7470 . 2 (𝒫 𝐴 ∈ V → 𝐴 ∈ V)
31, 2impbii 212 1 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∈ wcel 2115  Vcvv 3479  𝒫 cpw 4520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-pw 4522  df-sn 4549  df-pr 4551  df-uni 4820 This theorem is referenced by:  2pwuninel  8656  ranklim  9257  r1pwALT  9259  isf34lem6  9787  isfin1-2  9792  pwfseqlem4  10069  pwfseqlem5  10070  gchpwdom  10077  hargch  10080  numufl  22509
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