Theorem pwexb 7767

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 5351	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2		pwexr 7766	. 2 ⊢ (𝒫 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 212	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2146  Vcvv 3457  𝒫 cpw 4564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-ss 3923  df-pw 4566  df-sn 4592  df-pr 4594  df-uni 4875

This theorem is used by:  2pwuninel  9123  ranklim  9819  r1pwALT  9821  isf34lem6  10375  isfin1-2  10380  pwfseqlem4  10658  pwfseqlem5  10659  gchpwdom  10666  hargch  10669  numufl  24101