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Theorem pwv 3827
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 3199 . . . 4  |-  x  C_  _V
2 vex 2792 . . . . 5  |-  x  e. 
_V
32elpw 3632 . . . 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 2281 1  |-  ~P _V  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1628    e. wcel 1688   _Vcvv 2789    C_ wss 3153   ~Pcpw 3626
This theorem is referenced by:  univ  4564  ncanth  6288
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791  df-in 3160  df-ss 3167  df-pw 3628
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