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Theorem pwundif 4760
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 3854 . 2  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  =  ( ~P ( A  u.  B )  u.  ~P A )
2 pwunss 4758 . . . . 5  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
3 unss 3620 . . . . 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 214 . . . 4  |-  ( ~P A  C_  ~P ( A  u.  B )  /\  ~P B  C_  ~P ( A  u.  B
) )
54simpli 464 . . 3  |-  ~P A  C_ 
~P ( A  u.  B )
6 ssequn2 3619 . . 3  |-  ( ~P A  C_  ~P ( A  u.  B )  <->  ( ~P ( A  u.  B )  u.  ~P A )  =  ~P ( A  u.  B
) )
75, 6mpbi 213 . 2  |-  ( ~P ( A  u.  B
)  u.  ~P A
)  =  ~P ( A  u.  B )
81, 7eqtr2i 2485 1  |-  ~P ( A  u.  B )  =  ( ( ~P ( A  u.  B
)  \  ~P A
)  u.  ~P A
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 375    = wceq 1455    \ cdif 3413    u. cun 3414    C_ wss 3416   ~Pcpw 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-pw 3965
This theorem is referenced by:  pwfilem  7894
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