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Theorem pwel 4227
Description: Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
Assertion
Ref Expression
pwel  |-  ( A  e.  B  ->  ~P A  e.  ~P ~P U. B )

Proof of Theorem pwel
StepHypRef Expression
1 elssuni 3857 . . 3  |-  ( A  e.  B  ->  A  C_ 
U. B )
2 sspwb 4225 . . 3  |-  ( A 
C_  U. B  <->  ~P A  C_ 
~P U. B )
31, 2sylib 188 . 2  |-  ( A  e.  B  ->  ~P A  C_  ~P U. B
)
4 pwexg 4196 . . 3  |-  ( A  e.  B  ->  ~P A  e.  _V )
5 elpwg 3634 . . 3  |-  ( ~P A  e.  _V  ->  ( ~P A  e.  ~P ~P U. B  <->  ~P A  C_ 
~P U. B ) )
64, 5syl 15 . 2  |-  ( A  e.  B  ->  ( ~P A  e.  ~P ~P U. B  <->  ~P A  C_ 
~P U. B ) )
73, 6mpbird 223 1  |-  ( A  e.  B  ->  ~P A  e.  ~P ~P U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1686   _Vcvv 2790    C_ wss 3154   ~Pcpw 3627   U.cuni 3829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-pw 3629  df-sn 3648  df-pr 3649  df-uni 3830
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