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Theorem pwv 3647
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 3044 . . . 4  |-  x  C_  _V
2 vex 2622 . . . . 5  |-  x  e. 
_V
32elpw 3431 . . . 4  |-  ( x  e.  ~P _V  <->  x  C_  _V )
41, 3mpbir 144 . . 3  |-  x  e. 
~P _V
54, 22th 172 . 2  |-  ( x  e.  ~P _V  <->  x  e.  _V )
65eqriv 2085 1  |-  ~P _V  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   _Vcvv 2619    C_ wss 2997   ~Pcpw 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427
This theorem is referenced by:  univ  4288
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