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Theorem pwundifss 4300
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 3512 . 2  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  C_  ( ~P ( A  u.  B
)  u.  ~P A
)
2 pwunss 4298 . . . . 5  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
3 unss 3324 . . . . 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 146 . . . 4  |-  ( ~P A  C_  ~P ( A  u.  B )  /\  ~P B  C_  ~P ( A  u.  B
) )
54simpli 111 . . 3  |-  ~P A  C_ 
~P ( A  u.  B )
6 ssequn2 3323 . . 3  |-  ( ~P A  C_  ~P ( A  u.  B )  <->  ( ~P ( A  u.  B )  u.  ~P A )  =  ~P ( A  u.  B
) )
75, 6mpbi 145 . 2  |-  ( ~P ( A  u.  B
)  u.  ~P A
)  =  ~P ( A  u.  B )
81, 7sseqtri 3204 1  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  C_  ~P ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1364    \ cdif 3141    u. cun 3142    C_ wss 3144   ~Pcpw 3590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592
This theorem is referenced by: (None)
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