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Theorem pwunss 4726
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 3635 . . 3  |-  ( ( x  C_  A  \/  x  C_  B )  ->  x  C_  ( A  u.  B ) )
2 elun 3597 . . . 4  |-  ( x  e.  ( ~P A  u.  ~P B )  <->  ( x  e.  ~P A  \/  x  e.  ~P B ) )
3 selpw 3967 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
4 selpw 3967 . . . . 5  |-  ( x  e.  ~P B  <->  x  C_  B
)
53, 4orbi12i 521 . . . 4  |-  ( ( x  e.  ~P A  \/  x  e.  ~P B )  <->  ( x  C_  A  \/  x  C_  B ) )
62, 5bitri 249 . . 3  |-  ( x  e.  ( ~P A  u.  ~P B )  <->  ( x  C_  A  \/  x  C_  B ) )
7 selpw 3967 . . 3  |-  ( x  e.  ~P ( A  u.  B )  <->  x  C_  ( A  u.  B )
)
81, 6, 73imtr4i 266 . 2  |-  ( x  e.  ( ~P A  u.  ~P B )  ->  x  e.  ~P ( A  u.  B )
)
98ssriv 3460 1  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    e. wcel 1758    u. cun 3426    C_ wss 3428   ~Pcpw 3960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3072  df-un 3433  df-in 3435  df-ss 3442  df-pw 3962
This theorem is referenced by:  pwundif  4728  pwun  4729
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