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Theorem pwin 4400
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 3479 . . . 4  |-  ( ( x  C_  A  /\  x  C_  B )  <->  x  C_  ( A  i^i  B ) )
2 vex 2876 . . . . . 6  |-  x  e. 
_V
32elpw 3720 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
42elpw 3720 . . . . 5  |-  ( x  e.  ~P B  <->  x  C_  B
)
53, 4anbi12i 678 . . . 4  |-  ( ( x  e.  ~P A  /\  x  e.  ~P B )  <->  ( x  C_  A  /\  x  C_  B ) )
62elpw 3720 . . . 4  |-  ( x  e.  ~P ( A  i^i  B )  <->  x  C_  ( A  i^i  B ) )
71, 5, 63bitr4i 268 . . 3  |-  ( ( x  e.  ~P A  /\  x  e.  ~P B )  <->  x  e.  ~P ( A  i^i  B
) )
87ineqri 3450 . 2  |-  ( ~P A  i^i  ~P B
)  =  ~P ( A  i^i  B )
98eqcomi 2370 1  |-  ~P ( A  i^i  B )  =  ( ~P A  i^i  ~P B )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1647    e. wcel 1715    i^i cin 3237    C_ wss 3238   ~Pcpw 3714
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-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
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-v 2875  df-in 3245  df-ss 3252  df-pw 3716
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