Theorem pwexb 7782

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 5339	. 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2		pwexr 7781	. 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 2144  Vcvv 3450  𝒫 cpw 4556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-un 4150  df-pw 4558  df-sn 4585  df-pr 4587  df-uni 4867

This theorem is used by:  2pwuninel  9156  ranklim  9863  r1pwALT  9865  isf34lem6  10463  isfin1-2  10468  pwfseqlem4  10752  pwfseqlem5  10753  gchpwdom  10760  hargch  10763  numufl  24241