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Theorem pwel 4375
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 4003 . . 3  |-  ( A  e.  B  ->  A  C_ 
U. B )
2 sspwb 4373 . . 3  |-  ( A 
C_  U. B  <->  ~P A  C_ 
~P U. B )
31, 2sylib 189 . 2  |-  ( A  e.  B  ->  ~P A  C_  ~P U. B
)
4 pwexg 4343 . . 3  |-  ( A  e.  B  ->  ~P A  e.  _V )
5 elpwg 3766 . . 3  |-  ( ~P A  e.  _V  ->  ( ~P A  e.  ~P ~P U. B  <->  ~P A  C_ 
~P U. B ) )
64, 5syl 16 . 2  |-  ( A  e.  B  ->  ( ~P A  e.  ~P ~P U. B  <->  ~P A  C_ 
~P U. B ) )
73, 6mpbird 224 1  |-  ( A  e.  B  ->  ~P A  e.  ~P ~P U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   U.cuni 3975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-pw 3761  df-sn 3780  df-pr 3781  df-uni 3976
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