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Theorem pwexb 7787
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 5377 . 2 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2 pwexr 7786 . 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 2107  Vcvv 3479  𝒫 cpw 4599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955  df-ss 3967  df-pw 4601  df-sn 4626  df-pr 4628  df-uni 4907
This theorem is referenced by:  2pwuninel  9173  ranklim  9885  r1pwALT  9887  isf34lem6  10421  isfin1-2  10426  pwfseqlem4  10703  pwfseqlem5  10704  gchpwdom  10711  hargch  10714  numufl  23924
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