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Theorem pwin 4726
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 3660 . . . 4  |-  ( ( x  C_  A  /\  x  C_  B )  <->  x  C_  ( A  i^i  B ) )
2 selpw 3961 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
3 selpw 3961 . . . . 5  |-  ( x  e.  ~P B  <->  x  C_  B
)
42, 3anbi12i 695 . . . 4  |-  ( ( x  e.  ~P A  /\  x  e.  ~P B )  <->  ( x  C_  A  /\  x  C_  B ) )
5 selpw 3961 . . . 4  |-  ( x  e.  ~P ( A  i^i  B )  <->  x  C_  ( A  i^i  B ) )
61, 4, 53bitr4i 277 . . 3  |-  ( ( x  e.  ~P A  /\  x  e.  ~P B )  <->  x  e.  ~P ( A  i^i  B
) )
76ineqri 3632 . 2  |-  ( ~P A  i^i  ~P B
)  =  ~P ( A  i^i  B )
87eqcomi 2415 1  |-  ~P ( A  i^i  B )  =  ( ~P A  i^i  ~P B )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1405    e. wcel 1842    i^i cin 3412    C_ wss 3413   ~Pcpw 3954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-in 3420  df-ss 3427  df-pw 3956
This theorem is referenced by: (None)
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