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Theorem sspwimpALT 34126
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 34126 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 33740 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 33731). 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 4008) . (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 3109 . . . . . . . 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 4008 . . . . . . . . 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 3503 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  C_  B
)
82, 7elpwgded 33731 . . . . . 6  |-  ( ( T.  /\  ( A 
C_  B  /\  x  e.  ~P A ) )  ->  x  e.  ~P B )
98uunT1 33971 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  e.  ~P B )
109ex 432 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
1110alrimiv 1724 . . 3  |-  ( A 
C_  B  ->  A. x
( x  e.  ~P A  ->  x  e.  ~P B ) )
12 dfss2 3478 . . . 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 33605 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1396   T. wtru 1399    e. wcel 1823   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-in 3468  df-ss 3475  df-pw 4001
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator