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Theorem pwel 4119
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 3753 . . 3  |-  ( A  e.  B  ->  A  C_ 
U. B )
2 sspwb 4117 . . 3  |-  ( A 
C_  U. B  <->  ~P A  C_ 
~P U. B )
31, 2sylib 190 . 2  |-  ( A  e.  B  ->  ~P A  C_  ~P U. B
)
4 pwexg 4088 . . 3  |-  ( A  e.  B  ->  ~P A  e.  _V )
5 elpwg 3537 . . 3  |-  ( ~P A  e.  _V  ->  ( ~P A  e.  ~P ~P U. B  <->  ~P A  C_ 
~P U. B ) )
64, 5syl 17 . 2  |-  ( A  e.  B  ->  ( ~P A  e.  ~P ~P U. B  <->  ~P A  C_ 
~P U. B ) )
73, 6mpbird 225 1  |-  ( A  e.  B  ->  ~P A  e.  ~P ~P U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    e. wcel 1621   _Vcvv 2727    C_ wss 3078   ~Pcpw 3530   U.cuni 3727
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-pw 3532  df-sn 3550  df-pr 3551  df-uni 3728
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