Theorem pwexb 7761

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 5347	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2		pwexr 7760	. 2 ⊢ (𝒫 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 212	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Syntax hints:   ↔ wb 209   ∈ wcel 2149  Vcvv 3463  𝒫 cpw 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-pw 4566  df-sn 4592  df-pr 4594  df-uni 4874

This theorem is referenced by:  2pwuninel  9116  ranklim  9812  r1pwALT  9814  isf34lem6  10360  isfin1-2  10365  pwfseqlem4  10643  pwfseqlem5  10644  gchpwdom  10651  hargch  10654  numufl  24037