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Theorem pwundif 4316
Description: Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.)
Assertion
Ref Expression
pwundif  |-  ~P ( A  u.  B )  =  ( ( ~P ( A  u.  B
)  \  ~P A
)  u.  ~P A
)

Proof of Theorem pwundif
StepHypRef Expression
1 undif1 3542 . 2  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  =  ( ~P ( A  u.  B )  u.  ~P A )
2 pwunss 4314 . . . . 5  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
3 unss 3362 . . . . 5  |-  ( ( ~P A  C_  ~P ( A  u.  B
)  /\  ~P B  C_ 
~P ( A  u.  B ) )  <->  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
)
42, 3mpbir 200 . . . 4  |-  ( ~P A  C_  ~P ( A  u.  B )  /\  ~P B  C_  ~P ( A  u.  B
) )
54simpli 444 . . 3  |-  ~P A  C_ 
~P ( A  u.  B )
6 ssequn2 3361 . . 3  |-  ( ~P A  C_  ~P ( A  u.  B )  <->  ( ~P ( A  u.  B )  u.  ~P A )  =  ~P ( A  u.  B
) )
75, 6mpbi 199 . 2  |-  ( ~P ( A  u.  B
)  u.  ~P A
)  =  ~P ( A  u.  B )
81, 7eqtr2i 2317 1  |-  ~P ( A  u.  B )  =  ( ( ~P ( A  u.  B
)  \  ~P A
)  u.  ~P A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    \ cdif 3162    u. cun 3163    C_ wss 3165   ~Pcpw 3638
This theorem is referenced by:  pwfilem  7166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640
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