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Theorem pwundifss 4270
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 3489 . 2  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  C_  ( ~P ( A  u.  B
)  u.  ~P A
)
2 pwunss 4268 . . . . 5  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
3 unss 3301 . . . . 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 145 . . . 4  |-  ( ~P A  C_  ~P ( A  u.  B )  /\  ~P B  C_  ~P ( A  u.  B
) )
54simpli 110 . . 3  |-  ~P A  C_ 
~P ( A  u.  B )
6 ssequn2 3300 . . 3  |-  ( ~P A  C_  ~P ( A  u.  B )  <->  ( ~P ( A  u.  B )  u.  ~P A )  =  ~P ( A  u.  B
) )
75, 6mpbi 144 . 2  |-  ( ~P ( A  u.  B
)  u.  ~P A
)  =  ~P ( A  u.  B )
81, 7sseqtri 3181 1  |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  C_  ~P ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348    \ cdif 3118    u. cun 3119    C_ wss 3121   ~Pcpw 3566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568
This theorem is referenced by: (None)
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