MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwundif Structured version   Unicode version

Theorem pwundif 4482
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 3695 . 2  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  =  ( ~P ( A  u.  B )  u.  ~P A )
2 pwunss 4480 . . . . 5  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
3 unss 3513 . . . . 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 201 . . . 4  |-  ( ~P A  C_  ~P ( A  u.  B )  /\  ~P B  C_  ~P ( A  u.  B
) )
54simpli 445 . . 3  |-  ~P A  C_ 
~P ( A  u.  B )
6 ssequn2 3512 . . 3  |-  ( ~P A  C_  ~P ( A  u.  B )  <->  ( ~P ( A  u.  B )  u.  ~P A )  =  ~P ( A  u.  B
) )
75, 6mpbi 200 . 2  |-  ( ~P ( A  u.  B
)  u.  ~P A
)  =  ~P ( A  u.  B )
81, 7eqtr2i 2456 1  |-  ~P ( A  u.  B )  =  ( ( ~P ( A  u.  B
)  \  ~P A
)  u.  ~P A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    \ cdif 3309    u. cun 3310    C_ wss 3312   ~Pcpw 3791
This theorem is referenced by:  pwfilem  7393
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-ne 2600  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-pw 3793
  Copyright terms: Public domain W3C validator