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Theorem pwv 3842
Description: The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
Assertion
Ref Expression
pwv  |-  ~P _V  =  _V

Proof of Theorem pwv
StepHypRef Expression
1 ssv 3211 . . . 4  |-  x  C_  _V
2 vex 2804 . . . . 5  |-  x  e. 
_V
32elpw 3644 . . . 4  |-  ( x  e.  ~P _V  <->  x  C_  _V )
41, 3mpbir 200 . . 3  |-  x  e. 
~P _V
54, 22th 230 . 2  |-  ( x  e.  ~P _V  <->  x  e.  _V )
65eqriv 2293 1  |-  ~P _V  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638
This theorem is referenced by:  univ  4581  ncanth  6311
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640
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