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Theorem pwunim 4069
Description: The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by Jim Kingdon, 30-Sep-2018.)
Assertion
Ref Expression
pwunim  |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B
)  =  ( ~P A  u.  ~P B
) )

Proof of Theorem pwunim
StepHypRef Expression
1 pwssunim 4067 . . 3  |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B
)  C_  ( ~P A  u.  ~P B
) )
2 pwunss 4066 . . . 4  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
32biantru 296 . . 3  |-  ( ~P ( A  u.  B
)  C_  ( ~P A  u.  ~P B
)  <->  ( ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B )  /\  ( ~P A  u.  ~P B )  C_  ~P ( A  u.  B
) ) )
41, 3sylib 120 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  -> 
( ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B )  /\  ( ~P A  u.  ~P B )  C_  ~P ( A  u.  B
) ) )
5 eqss 3023 . 2  |-  ( ~P ( A  u.  B
)  =  ( ~P A  u.  ~P B
)  <->  ( ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B )  /\  ( ~P A  u.  ~P B )  C_  ~P ( A  u.  B
) ) )
64, 5sylibr 132 1  |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B
)  =  ( ~P A  u.  ~P B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662    = wceq 1285    u. cun 2980    C_ wss 2982   ~Pcpw 3400
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-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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402
This theorem is referenced by: (None)
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