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Theorem pwin 4487
Description: The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
Assertion
Ref Expression
pwin  |-  ~P ( A  i^i  B )  =  ( ~P A  i^i  ~P B )

Proof of Theorem pwin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssin 3563 . . . 4  |-  ( ( x  C_  A  /\  x  C_  B )  <->  x  C_  ( A  i^i  B ) )
2 vex 2959 . . . . . 6  |-  x  e. 
_V
32elpw 3805 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
42elpw 3805 . . . . 5  |-  ( x  e.  ~P B  <->  x  C_  B
)
53, 4anbi12i 679 . . . 4  |-  ( ( x  e.  ~P A  /\  x  e.  ~P B )  <->  ( x  C_  A  /\  x  C_  B ) )
62elpw 3805 . . . 4  |-  ( x  e.  ~P ( A  i^i  B )  <->  x  C_  ( A  i^i  B ) )
71, 5, 63bitr4i 269 . . 3  |-  ( ( x  e.  ~P A  /\  x  e.  ~P B )  <->  x  e.  ~P ( A  i^i  B
) )
87ineqri 3534 . 2  |-  ( ~P A  i^i  ~P B
)  =  ~P ( A  i^i  B )
98eqcomi 2440 1  |-  ~P ( A  i^i  B )  =  ( ~P A  i^i  ~P B )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   ~Pcpw 3799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327  df-ss 3334  df-pw 3801
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