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Theorem sspwimpALT 28701
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 28701 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 28337 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 28330). 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 3633) . (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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . . . . 8  |-  x  e. 
_V
21a1i 10 . . . . . . 7  |-  (  T. 
->  x  e.  _V )
3 id 19 . . . . . . . . 9  |-  ( x  e.  ~P A  ->  x  e.  ~P A
)
4 elpwi 3633 . . . . . . . . 9  |-  ( x  e.  ~P A  ->  x  C_  A )
53, 4syl 15 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
6 id 19 . . . . . . . 8  |-  ( A 
C_  B  ->  A  C_  B )
75, 6sylan9ssr 3193 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  C_  B
)
82, 7elpwgded 28330 . . . . . 6  |-  ( (  T.  /\  ( A 
C_  B  /\  x  e.  ~P A ) )  ->  x  e.  ~P B )
98uunT1 28555 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  e.  ~P B )
109ex 423 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
1110alrimiv 1617 . . 3  |-  ( A 
C_  B  ->  A. x
( x  e.  ~P A  ->  x  e.  ~P B ) )
12 dfss2 3169 . . . 4  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
1312biimpri 197 . . 3  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  ->  ~P A  C_  ~P B
)
1411, 13syl 15 . 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 4    /\ wa 358    T. wtru 1307   A.wal 1527    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627
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