Theorem pwexb 7707

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

Step	Hyp	Ref	Expression
1		pwexg 5320	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2		pwexr 7706	. 2 ⊢ (𝒫 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 209	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Syntax hints:   ↔ wb 206   ∈ wcel 2113  Vcvv 3437  𝒫 cpw 4551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-un 3903  df-ss 3915  df-pw 4553  df-sn 4578  df-pr 4580  df-uni 4861

This theorem is referenced by:  2pwuninel  9054  ranklim  9746  r1pwALT  9748  isf34lem6  10280  isfin1-2  10285  pwfseqlem4  10562  pwfseqlem5  10563  gchpwdom  10570  hargch  10573  numufl  23833