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Theorem pwv 4006
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 3360 . . . 4  |-  x  C_  _V
2 vex 2951 . . . . 5  |-  x  e. 
_V
32elpw 3797 . . . 4  |-  ( x  e.  ~P _V  <->  x  C_  _V )
41, 3mpbir 201 . . 3  |-  x  e. 
~P _V
54, 22th 231 . 2  |-  ( x  e.  ~P _V  <->  x  e.  _V )
65eqriv 2432 1  |-  ~P _V  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791
This theorem is referenced by:  univ  4746  ncanth  6532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793
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