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Theorem pwv 3978
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 3332 . . . 4  |-  x  C_  _V
2 vex 2923 . . . . 5  |-  x  e. 
_V
32elpw 3769 . . . 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 2405 1  |-  ~P _V  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2920    C_ wss 3284   ~Pcpw 3763
This theorem is referenced by:  univ  4717  ncanth  6503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922  df-in 3291  df-ss 3298  df-pw 3765
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