MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwin Unicode version

Theorem pwin 4299
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 3393 . . . 4  |-  ( ( x  C_  A  /\  x  C_  B )  <->  x  C_  ( A  i^i  B ) )
2 vex 2793 . . . . . 6  |-  x  e. 
_V
32elpw 3633 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
42elpw 3633 . . . . 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 3633 . . . 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 3364 . 2  |-  ( ~P A  i^i  ~P B
)  =  ~P ( A  i^i  B )
98eqcomi 2289 1  |-  ~P ( A  i^i  B )  =  ( ~P A  i^i  ~P B )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1625    e. wcel 1686    i^i cin 3153    C_ wss 3154   ~Pcpw 3627
This theorem is referenced by:  selsubf  26001  selsubf3  26002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-in 3161  df-ss 3168  df-pw 3629
  Copyright terms: Public domain W3C validator