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Theorem pwunss 4744
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 3604 . . 3  |-  ( ( x  C_  A  \/  x  C_  B )  ->  x  C_  ( A  u.  B ) )
2 elun 3565 . . . 4  |-  ( x  e.  ( ~P A  u.  ~P B )  <->  ( x  e.  ~P A  \/  x  e.  ~P B ) )
3 selpw 3949 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
4 selpw 3949 . . . . 5  |-  ( x  e.  ~P B  <->  x  C_  B
)
53, 4orbi12i 530 . . . 4  |-  ( ( x  e.  ~P A  \/  x  e.  ~P B )  <->  ( x  C_  A  \/  x  C_  B ) )
62, 5bitri 257 . . 3  |-  ( x  e.  ( ~P A  u.  ~P B )  <->  ( x  C_  A  \/  x  C_  B ) )
7 selpw 3949 . . 3  |-  ( x  e.  ~P ( A  u.  B )  <->  x  C_  ( A  u.  B )
)
81, 6, 73imtr4i 274 . 2  |-  ( x  e.  ( ~P A  u.  ~P B )  ->  x  e.  ~P ( A  u.  B )
)
98ssriv 3422 1  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 375    e. wcel 1904    u. cun 3388    C_ wss 3390   ~Pcpw 3942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-un 3395  df-in 3397  df-ss 3404  df-pw 3944
This theorem is referenced by:  pwundif  4746  pwun  4747
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