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Theorem sspwimpcf 27965
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 27965, using conventional notation, was translated from its virtual deduction form, sspwimpcfVD 27966, 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 )
Dummy variable  x is distinct from all other variables.

Proof of Theorem sspwimpcf
StepHypRef Expression
1 vex 2793 . . . . . 6  |-  x  e. 
_V
2 id 21 . . . . . . 7  |-  ( A 
C_  B  ->  A  C_  B )
3 id 21 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  e.  ~P A
)
4 elpwi 3635 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
53, 4syl 17 . . . . . . 7  |-  ( x  e.  ~P A  ->  x  C_  A )
6 sstr2 3188 . . . . . . . 8  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
76impcom 421 . . . . . . 7  |-  ( ( A  C_  B  /\  x  C_  A )  ->  x  C_  B )
82, 5, 7syl2an 465 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  C_  B
)
9 elpwg 3634 . . . . . . 7  |-  ( x  e.  _V  ->  (
x  e.  ~P B  <->  x 
C_  B ) )
109biimpar 473 . . . . . 6  |-  ( ( x  e.  _V  /\  x  C_  B )  ->  x  e.  ~P B
)
111, 8, 10eel021old 27742 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  e.  ~P B )
1211ex 425 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
1312alrimiv 1618 . . 3  |-  ( A 
C_  B  ->  A. x
( x  e.  ~P A  ->  x  e.  ~P B ) )
14 dfss2 3171 . . . 4  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
1514biimpri 199 . . 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 27612 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1528    e. wcel 1685   _Vcvv 2790    C_ wss 3154   ~Pcpw 3627
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-in 3161  df-ss 3168  df-pw 3629
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