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

Theorem pwexb 6929
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 uniexb 6928 . 2 (𝒫 𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
2 unipw 4884 . . 3 𝒫 𝐴 = 𝐴
32eleq1i 2689 . 2 ( 𝒫 𝐴 ∈ V ↔ 𝐴 ∈ V)
41, 3bitr2i 265 1 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1987  Vcvv 3189  𝒫 cpw 4135   cuni 4407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-pw 4137  df-sn 4154  df-pr 4156  df-uni 4408
This theorem is referenced by:  pwuninel  7353  2pwuninel  8067  pwfi  8213  pwwf  8622  ranklim  8659  r1pw  8660  r1pwALT  8661  isfin3  9070  isf34lem6  9154  isfin1-2  9159  pwfseqlem4  9436  pwfseqlem5  9437  gchpwdom  9444  hargch  9447  dis2ndc  21186  numufl  21642
  Copyright terms: Public domain W3C validator