ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pwssunim Unicode version

Theorem pwssunim 4067
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 3155 . . . . 5  |-  ( B 
C_  A  <->  ( A  u.  B )  =  A )
2 pweq 3403 . . . . . 6  |-  ( ( A  u.  B )  =  A  ->  ~P ( A  u.  B
)  =  ~P A
)
3 eqimss 3060 . . . . . 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 3152 . . . . 5  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )
7 pweq 3403 . . . . . 6  |-  ( ( A  u.  B )  =  B  ->  ~P ( A  u.  B
)  =  ~P B
)
8 eqimss 3060 . . . . . 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 3161 . 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 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:  pwunim  4069
  Copyright terms: Public domain W3C validator