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Theorem sspwimpALT2 36772
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://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
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 3064 . . . 4  |-  x  e. 
_V
2 elpwi 3966 . . . . 5  |-  ( x  e.  ~P A  ->  x  C_  A )
3 id 23 . . . . 5  |-  ( A 
C_  B  ->  A  C_  B )
42, 3sylan9ssr 3458 . . . 4  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  C_  B
)
5 elpwg 3965 . . . . 5  |-  ( x  e.  _V  ->  (
x  e.  ~P B  <->  x 
C_  B ) )
65biimpar 485 . . . 4  |-  ( ( x  e.  _V  /\  x  C_  B )  ->  x  e.  ~P B
)
71, 4, 6sylancr 663 . . 3  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  e.  ~P B )
87ex 434 . 2  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
98ssrdv 3450 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1844   _Vcvv 3061    C_ wss 3416   ~Pcpw 3957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-v 3063  df-in 3423  df-ss 3430  df-pw 3959
This theorem is referenced by: (None)
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