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Theorem pwel 4224
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 3856 . . 3  |-  ( A  e.  B  ->  A  C_ 
U. B )
2 sspwb 4222 . . 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 4193 . . 3  |-  ( A  e.  B  ->  ~P A  e.  _V )
5 elpwg 3633 . . 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 1685   _Vcvv 2789    C_ wss 3153   ~Pcpw 3626   U.cuni 3828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-pw 3628  df-sn 3647  df-pr 3648  df-uni 3829
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