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Theorem pwin 4174
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 3268 . . . 4  |-  ( ( x  C_  A  /\  x  C_  B )  <->  x  C_  ( A  i^i  B ) )
2 vex 2663 . . . . . 6  |-  x  e. 
_V
32elpw 3486 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
42elpw 3486 . . . . 5  |-  ( x  e.  ~P B  <->  x  C_  B
)
53, 4anbi12i 455 . . . 4  |-  ( ( x  e.  ~P A  /\  x  e.  ~P B )  <->  ( x  C_  A  /\  x  C_  B ) )
62elpw 3486 . . . 4  |-  ( x  e.  ~P ( A  i^i  B )  <->  x  C_  ( A  i^i  B ) )
71, 5, 63bitr4i 211 . . 3  |-  ( ( x  e.  ~P A  /\  x  e.  ~P B )  <->  x  e.  ~P ( A  i^i  B
) )
87ineqri 3239 . 2  |-  ( ~P A  i^i  ~P B
)  =  ~P ( A  i^i  B )
98eqcomi 2121 1  |-  ~P ( A  i^i  B )  =  ( ~P A  i^i  ~P B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316    e. wcel 1465    i^i cin 3040    C_ wss 3041   ~Pcpw 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482
This theorem is referenced by: (None)
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