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Theorem sscon 3224
Description: Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
sscon  |-  ( A 
C_  B  ->  ( C  \  B )  C_  ( C  \  A ) )

Proof of Theorem sscon
StepHypRef Expression
1 ssel 3097 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21con3d 127 . . . 4  |-  ( A 
C_  B  ->  ( -.  x  e.  B  ->  -.  x  e.  A
) )
32anim2d 550 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  C  /\  -.  x  e.  B
)  ->  ( x  e.  C  /\  -.  x  e.  A ) ) )
4 eldif 3088 . . 3  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
5 eldif 3088 . . 3  |-  ( x  e.  ( C  \  A )  <->  ( x  e.  C  /\  -.  x  e.  A ) )
63, 4, 53imtr4g 263 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( C 
\  B )  ->  x  e.  ( C  \  A ) ) )
76ssrdv 3106 1  |-  ( A 
C_  B  ->  ( C  \  B )  C_  ( C  \  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    e. wcel 1621    \ cdif 3075    C_ wss 3078
This theorem is referenced by:  sorpsscmpl  6140  sbthlem1  6856  sbthlem2  6857  cantnfp1lem1  7264  cantnfp1lem3  7266  mapfien  7283  fin23lem26  7835  isf34lem7  7889  isf34lem6  7890  isercoll2  12019  setsres  13048  mplsubglem  16011  fctop  16573  cctop  16575  clsval2  16619  ntrss  16624  hauscmplem  16965  iunconlem  16985  clscon  16988  ptbasin  17104  regr1lem  17262  cfinfil  17420  csdfil  17421  blcld  17883  voliunlem1  18739  uniioombllem5  18774  kur14lem6  22913
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-dif 3081  df-in 3085  df-ss 3089
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