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Theorem pwunss 4243
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 3286 . . 3  |-  ( ( x  C_  A  \/  x  C_  B )  ->  x  C_  ( A  u.  B ) )
2 elun 3248 . . . 4  |-  ( x  e.  ( ~P A  u.  ~P B )  <->  ( x  e.  ~P A  \/  x  e.  ~P B ) )
3 vex 2715 . . . . . 6  |-  x  e. 
_V
43elpw 3549 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3549 . . . . 5  |-  ( x  e.  ~P B  <->  x  C_  B
)
64, 5orbi12i 754 . . . 4  |-  ( ( x  e.  ~P A  \/  x  e.  ~P B )  <->  ( x  C_  A  \/  x  C_  B ) )
72, 6bitri 183 . . 3  |-  ( x  e.  ( ~P A  u.  ~P B )  <->  ( x  C_  A  \/  x  C_  B ) )
83elpw 3549 . . 3  |-  ( x  e.  ~P ( A  u.  B )  <->  x  C_  ( A  u.  B )
)
91, 7, 83imtr4i 200 . 2  |-  ( x  e.  ( ~P A  u.  ~P B )  ->  x  e.  ~P ( A  u.  B )
)
109ssriv 3132 1  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
Colors of variables: wff set class
Syntax hints:    \/ wo 698    e. wcel 2128    u. cun 3100    C_ wss 3102   ~Pcpw 3543
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545
This theorem is referenced by:  pwundifss  4245  pwunim  4246
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