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Theorem pwv 4189
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 3438 . . . 4  |-  x  C_  _V
2 selpw 3949 . . . 4  |-  ( x  e.  ~P _V  <->  x  C_  _V )
31, 2mpbir 214 . . 3  |-  x  e. 
~P _V
4 vex 3034 . . 3  |-  x  e. 
_V
53, 42th 247 . 2  |-  ( x  e.  ~P _V  <->  x  e.  _V )
65eqriv 2468 1  |-  ~P _V  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    e. wcel 1904   _Vcvv 3031    C_ wss 3390   ~Pcpw 3942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-in 3397  df-ss 3404  df-pw 3944
This theorem is referenced by:  univ  4651  ncanth  6268
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