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Theorem pwundif 4300
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 3529 . 2  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  =  ( ~P ( A  u.  B )  u.  ~P A )
2 pwunss 4298 . . . . 5  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
3 unss 3349 . . . . 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 3348 . . 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 2304 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 1623    \ cdif 3149    u. cun 3150    C_ wss 3152   ~Pcpw 3625
This theorem is referenced by:  pwfilem  7150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627
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