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Theorem pwundif 4730
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 3847 . 2  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  =  ( ~P ( A  u.  B )  u.  ~P A )
2 pwunss 4728 . . . . 5  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
3 unss 3617 . . . . 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 209 . . . 4  |-  ( ~P A  C_  ~P ( A  u.  B )  /\  ~P B  C_  ~P ( A  u.  B
) )
54simpli 456 . . 3  |-  ~P A  C_ 
~P ( A  u.  B )
6 ssequn2 3616 . . 3  |-  ( ~P A  C_  ~P ( A  u.  B )  <->  ( ~P ( A  u.  B )  u.  ~P A )  =  ~P ( A  u.  B
) )
75, 6mpbi 208 . 2  |-  ( ~P ( A  u.  B
)  u.  ~P A
)  =  ~P ( A  u.  B )
81, 7eqtr2i 2432 1  |-  ~P ( A  u.  B )  =  ( ( ~P ( A  u.  B
)  \  ~P A
)  u.  ~P A
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1405    \ cdif 3411    u. cun 3412    C_ wss 3414   ~Pcpw 3955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-pw 3957
This theorem is referenced by:  pwfilem  7848
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