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

Theorem sspwimpcf 34140
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. sspwimpcf 34140, using conventional notation, was translated from its virtual deduction form, sspwimpcfVD 34141, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimpcf  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )

Proof of Theorem sspwimpcf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 3109 . . . . . 6  |-  x  e. 
_V
2 id 22 . . . . . . 7  |-  ( A 
C_  B  ->  A  C_  B )
3 id 22 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  e.  ~P A
)
4 elpwi 4008 . . . . . . . 8  |-  ( x  e.  ~P A  ->  x  C_  A )
53, 4syl 16 . . . . . . 7  |-  ( x  e.  ~P A  ->  x  C_  A )
6 sstr2 3496 . . . . . . . 8  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
76impcom 428 . . . . . . 7  |-  ( ( A  C_  B  /\  x  C_  A )  ->  x  C_  B )
82, 5, 7syl2an 475 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  C_  B
)
9 elpwg 4007 . . . . . . 7  |-  ( x  e.  _V  ->  (
x  e.  ~P B  <->  x 
C_  B ) )
109biimpar 483 . . . . . 6  |-  ( ( x  e.  _V  /\  x  C_  B )  ->  x  e.  ~P B
)
111, 8, 10eel021old 33899 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  ~P A
)  ->  x  e.  ~P B )
1211ex 432 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  ~P A  ->  x  e.  ~P B
) )
1312alrimiv 1724 . . 3  |-  ( A 
C_  B  ->  A. x
( x  e.  ~P A  ->  x  e.  ~P B ) )
14 dfss2 3478 . . . 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 33762 1  |-  ( A 
C_  B  ->  ~P A  C_  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1396    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-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