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Theorem sspwimpALT 31963
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 31963 is the completed proof in conventional notation of the Virtual Deduction proof http://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 31584 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 31575). 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 3969) . (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 3073 . . . . . . . 8  |-  x  e. 
_V
21a1i 11 . . . . . . 7  |-  ( T. 
->  x  e.  _V )
3 id 22 . . . . . . . . 9  |-  ( x  e.  ~P A  ->  x  e.  ~P A
)
4 elpwi 3969 . . . . . . . . 9  |-  ( x  e.  ~P A  ->  x  C_  A )
53, 4syl 16 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
6 id 22 . . . . . . . 8  |-  ( A 
C_  B  ->  A  C_  B )
75, 6sylan9ssr 3470 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  C_  B
)
82, 7elpwgded 31575 . . . . . 6  |-  ( ( T.  /\  ( A 
C_  B  /\  x  e.  ~P A ) )  ->  x  e.  ~P B )
98uunT1 31815 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  e.  ~P B )
109ex 434 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
1110alrimiv 1686 . . 3  |-  ( A 
C_  B  ->  A. x
( x  e.  ~P A  ->  x  e.  ~P B ) )
12 dfss2 3445 . . . 4  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
1312biimpri 206 . . 3  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  ->  ~P A  C_  ~P B
)
1411, 13syl 16 . 2  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
1514idiALT 31455 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368   T. wtru 1371    e. wcel 1758   _Vcvv 3070    C_ wss 3428   ~Pcpw 3960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3072  df-in 3435  df-ss 3442  df-pw 3962
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator