Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sspwimp Structured version   Unicode version

Theorem sspwimp 33861
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. sspwimp 33861, using conventional notation, was translated from virtual deduction form, sspwimpVD 33862, using a translation program. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimp  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )

Proof of Theorem sspwimp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . . . . 7  |-  x  e. 
_V
21a1i 11 . . . . . 6  |-  ( T. 
->  x  e.  _V )
3 id 22 . . . . . . 7  |-  ( A 
C_  B  ->  A  C_  B )
4 id 22 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  e.  ~P A
)
5 elpwi 4024 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
64, 5syl 16 . . . . . . 7  |-  ( x  e.  ~P A  ->  x  C_  A )
7 sstr 3507 . . . . . . . 8  |-  ( ( x  C_  A  /\  A  C_  B )  ->  x  C_  B )
87ancoms 453 . . . . . . 7  |-  ( ( A  C_  B  /\  x  C_  A )  ->  x  C_  B )
93, 6, 8syl2an 477 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  C_  B
)
102, 9elpwgded 33480 . . . . . 6  |-  ( ( T.  /\  ( A 
C_  B  /\  x  e.  ~P A ) )  ->  x  e.  ~P B )
112, 9, 10uun0.1 33718 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  e.  ~P B )
1211ex 434 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
1312alrimiv 1720 . . 3  |-  ( A 
C_  B  ->  A. x
( x  e.  ~P A  ->  x  e.  ~P B ) )
14 dfss2 3488 . . . 4  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
1514biimpri 206 . . 3  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  ->  ~P A  C_  ~P B
)
1613, 15syl 16 . 2  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
1716iin1 33492 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1393   T. wtru 1396    e. wcel 1819   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-in 3478  df-ss 3485  df-pw 4017
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator