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Theorem pwundifss 4048
Description: Break up the power class of a union into a union of smaller classes. (Contributed by Jim Kingdon, 30-Sep-2018.)
Assertion
Ref Expression
pwundifss  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  C_  ~P ( A  u.  B
)

Proof of Theorem pwundifss
StepHypRef Expression
1 undif1ss 3325 . 2  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  C_  ( ~P ( A  u.  B
)  u.  ~P A
)
2 pwunss 4046 . . . . 5  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
3 unss 3147 . . . . 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 144 . . . 4  |-  ( ~P A  C_  ~P ( A  u.  B )  /\  ~P B  C_  ~P ( A  u.  B
) )
54simpli 109 . . 3  |-  ~P A  C_ 
~P ( A  u.  B )
6 ssequn2 3146 . . 3  |-  ( ~P A  C_  ~P ( A  u.  B )  <->  ( ~P ( A  u.  B )  u.  ~P A )  =  ~P ( A  u.  B
) )
75, 6mpbi 143 . 2  |-  ( ~P ( A  u.  B
)  u.  ~P A
)  =  ~P ( A  u.  B )
81, 7sseqtri 3032 1  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  C_  ~P ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    \ cdif 2971    u. cun 2972    C_ wss 2974   ~Pcpw 3390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392
This theorem is referenced by: (None)
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