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Theorem pwv 3607
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 2993 . . . 4  |-  x  C_  _V
2 vex 2577 . . . . 5  |-  x  e. 
_V
32elpw 3393 . . . 4  |-  ( x  e.  ~P _V  <->  x  C_  _V )
41, 3mpbir 138 . . 3  |-  x  e. 
~P _V
54, 22th 167 . 2  |-  ( x  e.  ~P _V  <->  x  e.  _V )
65eqriv 2053 1  |-  ~P _V  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409   _Vcvv 2574    C_ wss 2945   ~Pcpw 3387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389
This theorem is referenced by:  univ  4235
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