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Theorem sspwimpcf 29012
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. sspwimpcf 29012, using conventional notation, was translated from its virtual deduction form, sspwimpcfVD 29013, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimpcf  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )

Proof of Theorem sspwimpcf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2804 . . . . . 6  |-  x  e. 
_V
2 id 19 . . . . . . 7  |-  ( A 
C_  B  ->  A  C_  B )
3 id 19 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  e.  ~P A
)
4 elpwi 3646 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
53, 4syl 15 . . . . . . 7  |-  ( x  e.  ~P A  ->  x  C_  A )
6 sstr2 3199 . . . . . . . 8  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
76impcom 419 . . . . . . 7  |-  ( ( A  C_  B  /\  x  C_  A )  ->  x  C_  B )
82, 5, 7syl2an 463 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  C_  B
)
9 elpwg 3645 . . . . . . 7  |-  ( x  e.  _V  ->  (
x  e.  ~P B  <->  x 
C_  B ) )
109biimpar 471 . . . . . 6  |-  ( ( x  e.  _V  /\  x  C_  B )  ->  x  e.  ~P B
)
111, 8, 10eel021old 28778 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  e.  ~P B )
1211ex 423 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
1312alrimiv 1621 . . 3  |-  ( A 
C_  B  ->  A. x
( x  e.  ~P A  ->  x  e.  ~P B ) )
14 dfss2 3182 . . . 4  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
1514biimpri 197 . . 3  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  ->  ~P A  C_  ~P B
)
1613, 15syl 15 . 2  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
1716iin1 28639 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640
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