MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwexb Structured version   Visualization version   GIF version

Theorem pwexb 7132
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
StepHypRef Expression
1 pwexg 4991 . 2 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2 pwexr 7131 . 2 (𝒫 𝐴 ∈ V → 𝐴 ∈ V)
31, 2impbii 199 1 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2131  Vcvv 3332  𝒫 cpw 4294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-rex 3048  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-pw 4296  df-sn 4314  df-pr 4316  df-uni 4581
This theorem is referenced by:  2pwuninel  8272  ranklim  8872  r1pwALT  8874  isf34lem6  9386  isfin1-2  9391  pwfseqlem4  9668  pwfseqlem5  9669  gchpwdom  9676  hargch  9679  numufl  21912
  Copyright terms: Public domain W3C validator