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Theorem pwunim 4208
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 4206 . . 3  |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B
)  C_  ( ~P A  u.  ~P B
) )
2 pwunss 4205 . . . 4  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
32biantru 300 . . 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 121 . 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 3112 . 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 133 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 103    \/ wo 697    = wceq 1331    u. cun 3069    C_ wss 3071   ~Pcpw 3510
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512
This theorem is referenced by: (None)
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