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Theorem pwunss 4480
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 3518 . . 3  |-  ( ( x  C_  A  \/  x  C_  B )  ->  x  C_  ( A  u.  B ) )
2 elun 3480 . . . 4  |-  ( x  e.  ( ~P A  u.  ~P B )  <->  ( x  e.  ~P A  \/  x  e.  ~P B ) )
3 vex 2951 . . . . . 6  |-  x  e. 
_V
43elpw 3797 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3797 . . . . 5  |-  ( x  e.  ~P B  <->  x  C_  B
)
64, 5orbi12i 508 . . . 4  |-  ( ( x  e.  ~P A  \/  x  e.  ~P B )  <->  ( x  C_  A  \/  x  C_  B ) )
72, 6bitri 241 . . 3  |-  ( x  e.  ( ~P A  u.  ~P B )  <->  ( x  C_  A  \/  x  C_  B ) )
83elpw 3797 . . 3  |-  ( x  e.  ~P ( A  u.  B )  <->  x  C_  ( A  u.  B )
)
91, 7, 83imtr4i 258 . 2  |-  ( x  e.  ( ~P A  u.  ~P B )  ->  x  e.  ~P ( A  u.  B )
)
109ssriv 3344 1  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
Colors of variables: wff set class
Syntax hints:    \/ wo 358    e. wcel 1725    u. cun 3310    C_ wss 3312   ~Pcpw 3791
This theorem is referenced by:  pwundif  4482  pwun  4483
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319  df-ss 3326  df-pw 3793
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