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Theorem sspwimpALT 27751
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. sspwimpALT 27751 is the completed proof in conventional notation of the Virtual Deduction proof http://www.virtualdeduction.com/sspwimpaltvd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 27390 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction ( e.g. , the sub-theorem whose assertion is step 9 used elpwgded 27383). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem ( e.g. , the sub-theorem whose assertion is step 5 used elpwi 3607) . (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimpALT  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )

Proof of Theorem sspwimpALT
StepHypRef Expression
1 vex 2766 . . . . . . . 8  |-  x  e. 
_V
21a1i 12 . . . . . . 7  |-  (  T. 
->  x  e.  _V )
3 id 21 . . . . . . . . 9  |-  ( x  e.  ~P A  ->  x  e.  ~P A
)
4 elpwi 3607 . . . . . . . . 9  |-  ( x  e.  ~P A  ->  x  C_  A )
53, 4syl 17 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
6 id 21 . . . . . . . 8  |-  ( A 
C_  B  ->  A  C_  B )
75, 6sylan9ssr 3168 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  C_  B
)
82, 7elpwgded 27383 . . . . . 6  |-  ( (  T.  /\  ( A 
C_  B  /\  x  e.  ~P A ) )  ->  x  e.  ~P B )
98uunT1 27605 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  e.  ~P B )
109ex 425 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
1110alrimiv 2013 . . 3  |-  ( A 
C_  B  ->  A. x
( x  e.  ~P A  ->  x  e.  ~P B ) )
12 dfss2 3144 . . . 4  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
1312biimpri 199 . . 3  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  ->  ~P A  C_  ~P B
)
1411, 13syl 17 . 2  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
1514idi 2 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    T. wtru 1312   A.wal 1532    e. wcel 1621   _Vcvv 2763    C_ wss 3127   ~Pcpw 3599
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765  df-in 3134  df-ss 3141  df-pw 3601
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