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Theorem pwel 4328
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 3957 . . 3  |-  ( A  e.  B  ->  A  C_ 
U. B )
2 sspwb 4326 . . 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 4296 . . 3  |-  ( A  e.  B  ->  ~P A  e.  _V )
5 elpwg 3721 . . 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 1715   _Vcvv 2873    C_ wss 3238   ~Pcpw 3714   U.cuni 3929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-pw 3716  df-sn 3735  df-pr 3736  df-uni 3930
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