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Theorem pwexb 7705
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 5338 . 2 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2 pwexr 7704 . 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 2107  Vcvv 3448  𝒫 cpw 4565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-un 3920  df-in 3922  df-ss 3932  df-pw 4567  df-sn 4592  df-pr 4594  df-uni 4871
This theorem is referenced by:  2pwuninel  9083  ranklim  9787  r1pwALT  9789  isf34lem6  10323  isfin1-2  10328  pwfseqlem4  10605  pwfseqlem5  10606  gchpwdom  10613  hargch  10616  numufl  23282
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