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Theorem pwssunim 4174
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 3217 . . . . 5  |-  ( B 
C_  A  <->  ( A  u.  B )  =  A )
2 pweq 3481 . . . . . 6  |-  ( ( A  u.  B )  =  A  ->  ~P ( A  u.  B
)  =  ~P A
)
3 eqimss 3119 . . . . . 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 120 . . . 4  |-  ( B 
C_  A  ->  ~P ( A  u.  B
)  C_  ~P A
)
6 ssequn1 3214 . . . . 5  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )
7 pweq 3481 . . . . . 6  |-  ( ( A  u.  B )  =  B  ->  ~P ( A  u.  B
)  =  ~P B
)
8 eqimss 3119 . . . . . 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 120 . . . 4  |-  ( A 
C_  B  ->  ~P ( A  u.  B
)  C_  ~P B
)
115, 10orim12i 731 . . 3  |-  ( ( B  C_  A  \/  A  C_  B )  -> 
( ~P ( A  u.  B )  C_  ~P A  \/  ~P ( A  u.  B
)  C_  ~P B
) )
1211orcoms 702 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  -> 
( ~P ( A  u.  B )  C_  ~P A  \/  ~P ( A  u.  B
)  C_  ~P B
) )
13 ssun 3223 . 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 680    = wceq 1314    u. cun 3037    C_ wss 3039   ~Pcpw 3478
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480
This theorem is referenced by:  pwunim  4176
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