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Theorem sspwimpALT2 28375
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. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in http://www.virtualdeduction.com/sspwimpaltvd.html. (Contributed by Alan Sare, 11-Sep-2016.)
Assertion
Ref Expression
sspwimpALT2  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )

Proof of Theorem sspwimpALT2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2895 . . . 4  |-  x  e. 
_V
2 elpwi 3743 . . . . 5  |-  ( x  e.  ~P A  ->  x  C_  A )
3 id 20 . . . . 5  |-  ( A 
C_  B  ->  A  C_  B )
42, 3sylan9ssr 3298 . . . 4  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  C_  B
)
5 elpwg 3742 . . . . 5  |-  ( x  e.  _V  ->  (
x  e.  ~P B  <->  x 
C_  B ) )
65biimpar 472 . . . 4  |-  ( ( x  e.  _V  /\  x  C_  B )  ->  x  e.  ~P B
)
71, 4, 6sylancr 645 . . 3  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  e.  ~P B )
87ex 424 . 2  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
98ssrdv 3290 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   _Vcvv 2892    C_ wss 3256   ~Pcpw 3735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-in 3263  df-ss 3270  df-pw 3737
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