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Theorem pwv 3726
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 3119 . . . 4  |-  x  C_  _V
2 vex 2730 . . . . 5  |-  x  e. 
_V
32elpw 3536 . . . 4  |-  ( x  e.  ~P _V  <->  x  C_  _V )
41, 3mpbir 202 . . 3  |-  x  e. 
~P _V
54, 22th 232 . 2  |-  ( x  e.  ~P _V  <->  x  e.  _V )
65eqriv 2250 1  |-  ~P _V  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   _Vcvv 2727    C_ wss 3078   ~Pcpw 3530
This theorem is referenced by:  univ  4456  ncanth  6179
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-in 3085  df-ss 3089  df-pw 3532
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