Theorem pwexb 7774

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 5382	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2		pwexr 7773	. 2 ⊢ (𝒫 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 208	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Syntax hints:   ↔ wb 205   ∈ wcel 2098  Vcvv 3473  𝒫 cpw 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-un 3954  df-in 3956  df-ss 3966  df-pw 4608  df-sn 4633  df-pr 4635  df-uni 4913

This theorem is referenced by:  2pwuninel  9163  ranklim  9875  r1pwALT  9877  isf34lem6  10411  isfin1-2  10416  pwfseqlem4  10693  pwfseqlem5  10694  gchpwdom  10701  hargch  10704  numufl  23839