ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pwunss Unicode version

Theorem pwunss 4280
Description: The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
Assertion
Ref Expression
pwunss  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )

Proof of Theorem pwunss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssun 3314 . . 3  |-  ( ( x  C_  A  \/  x  C_  B )  ->  x  C_  ( A  u.  B ) )
2 elun 3276 . . . 4  |-  ( x  e.  ( ~P A  u.  ~P B )  <->  ( x  e.  ~P A  \/  x  e.  ~P B ) )
3 vex 2740 . . . . . 6  |-  x  e. 
_V
43elpw 3580 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3580 . . . . 5  |-  ( x  e.  ~P B  <->  x  C_  B
)
64, 5orbi12i 764 . . . 4  |-  ( ( x  e.  ~P A  \/  x  e.  ~P B )  <->  ( x  C_  A  \/  x  C_  B ) )
72, 6bitri 184 . . 3  |-  ( x  e.  ( ~P A  u.  ~P B )  <->  ( x  C_  A  \/  x  C_  B ) )
83elpw 3580 . . 3  |-  ( x  e.  ~P ( A  u.  B )  <->  x  C_  ( A  u.  B )
)
91, 7, 83imtr4i 201 . 2  |-  ( x  e.  ( ~P A  u.  ~P B )  ->  x  e.  ~P ( A  u.  B )
)
109ssriv 3159 1  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
Colors of variables: wff set class
Syntax hints:    \/ wo 708    e. wcel 2148    u. cun 3127    C_ wss 3129   ~Pcpw 3574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576
This theorem is referenced by:  pwundifss  4282  pwunim  4283
  Copyright terms: Public domain W3C validator