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Theorem pwssunim 4087
Description: The power class of the union of two classes is a subset of the union of their power classes, if one class is a subclass of the other. One direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by Jim Kingdon, 30-Sep-2018.)
Assertion
Ref Expression
pwssunim  |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B
)  C_  ( ~P A  u.  ~P B
) )

Proof of Theorem pwssunim
StepHypRef Expression
1 ssequn2 3162 . . . . 5  |-  ( B 
C_  A  <->  ( A  u.  B )  =  A )
2 pweq 3418 . . . . . 6  |-  ( ( A  u.  B )  =  A  ->  ~P ( A  u.  B
)  =  ~P A
)
3 eqimss 3067 . . . . . 6  |-  ( ~P ( A  u.  B
)  =  ~P A  ->  ~P ( A  u.  B )  C_  ~P A )
42, 3syl 14 . . . . 5  |-  ( ( A  u.  B )  =  A  ->  ~P ( A  u.  B
)  C_  ~P A
)
51, 4sylbi 119 . . . 4  |-  ( B 
C_  A  ->  ~P ( A  u.  B
)  C_  ~P A
)
6 ssequn1 3159 . . . . 5  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )
7 pweq 3418 . . . . . 6  |-  ( ( A  u.  B )  =  B  ->  ~P ( A  u.  B
)  =  ~P B
)
8 eqimss 3067 . . . . . 6  |-  ( ~P ( A  u.  B
)  =  ~P B  ->  ~P ( A  u.  B )  C_  ~P B )
97, 8syl 14 . . . . 5  |-  ( ( A  u.  B )  =  B  ->  ~P ( A  u.  B
)  C_  ~P B
)
106, 9sylbi 119 . . . 4  |-  ( A 
C_  B  ->  ~P ( A  u.  B
)  C_  ~P B
)
115, 10orim12i 709 . . 3  |-  ( ( B  C_  A  \/  A  C_  B )  -> 
( ~P ( A  u.  B )  C_  ~P A  \/  ~P ( A  u.  B
)  C_  ~P B
) )
1211orcoms 682 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  -> 
( ~P ( A  u.  B )  C_  ~P A  \/  ~P ( A  u.  B
)  C_  ~P B
) )
13 ssun 3168 . 2  |-  ( ( ~P ( A  u.  B )  C_  ~P A  \/  ~P ( A  u.  B )  C_ 
~P B )  ->  ~P ( A  u.  B
)  C_  ( ~P A  u.  ~P B
) )
1412, 13syl 14 1  |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B
)  C_  ( ~P A  u.  ~P B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    = wceq 1287    u. cun 2986    C_ wss 2988   ~Pcpw 3415
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417
This theorem is referenced by:  pwunim  4089
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