Theorem pwexb 7716

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 5314	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2		pwexr 7715	. 2 ⊢ (𝒫 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 210	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Syntax hints:   ↔ wb 207   ∈ wcel 2119  Vcvv 3432  𝒫 cpw 4536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-ss 3907  df-pw 4538  df-sn 4563  df-pr 4565  df-uni 4846

This theorem is referenced by:  2pwuninel  9067  ranklim  9766  r1pwALT  9768  isf34lem6  10300  isfin1-2  10305  pwfseqlem4  10583  pwfseqlem5  10584  gchpwdom  10591  hargch  10594  numufl  23905