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Theorem sspwimp 37315
Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. sspwimp 37315, using conventional notation, was translated from virtual deduction form, sspwimpVD 37316, using a translation program. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimp  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )

Proof of Theorem sspwimp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 3048 . . . . . . 7  |-  x  e. 
_V
21a1i 11 . . . . . 6  |-  ( T. 
->  x  e.  _V )
3 id 22 . . . . . . 7  |-  ( A 
C_  B  ->  A  C_  B )
4 id 22 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  e.  ~P A
)
5 elpwi 3960 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
64, 5syl 17 . . . . . . 7  |-  ( x  e.  ~P A  ->  x  C_  A )
7 sstr 3440 . . . . . . . 8  |-  ( ( x  C_  A  /\  A  C_  B )  ->  x  C_  B )
87ancoms 455 . . . . . . 7  |-  ( ( A  C_  B  /\  x  C_  A )  ->  x  C_  B )
93, 6, 8syl2an 480 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  C_  B
)
102, 9elpwgded 36931 . . . . . 6  |-  ( ( T.  /\  ( A 
C_  B  /\  x  e.  ~P A ) )  ->  x  e.  ~P B )
112, 9, 10uun0.1 37165 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  e.  ~P B )
1211ex 436 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
1312alrimiv 1773 . . 3  |-  ( A 
C_  B  ->  A. x
( x  e.  ~P A  ->  x  e.  ~P B ) )
14 dfss2 3421 . . . 4  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
1514biimpri 210 . . 3  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  ->  ~P A  C_  ~P B
)
1613, 15syl 17 . 2  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
1716iin1 36942 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371   A.wal 1442   T. wtru 1445    e. wcel 1887   _Vcvv 3045    C_ wss 3404   ~Pcpw 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-in 3411  df-ss 3418  df-pw 3953
This theorem is referenced by: (None)
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