Theorem pwexb 7713

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 5315	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2		pwexr 7712	. 2 ⊢ (𝒫 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 209	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Syntax hints:   ↔ wb 206   ∈ wcel 2114  Vcvv 3430  𝒫 cpw 4542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-un 3895  df-ss 3907  df-pw 4544  df-sn 4569  df-pr 4571  df-uni 4852

This theorem is referenced by:  2pwuninel  9063  ranklim  9759  r1pwALT  9761  isf34lem6  10293  isfin1-2  10298  pwfseqlem4  10576  pwfseqlem5  10577  gchpwdom  10584  hargch  10587  numufl  23890