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Theorem sscon 3425
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssel 3286 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21con3d 127 . . . 4  |-  ( A 
C_  B  ->  ( -.  x  e.  B  ->  -.  x  e.  A
) )
32anim2d 549 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  C  /\  -.  x  e.  B
)  ->  ( x  e.  C  /\  -.  x  e.  A ) ) )
4 eldif 3274 . . 3  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
5 eldif 3274 . . 3  |-  ( x  e.  ( C  \  A )  <->  ( x  e.  C  /\  -.  x  e.  A ) )
63, 4, 53imtr4g 262 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( C 
\  B )  ->  x  e.  ( C  \  A ) ) )
76ssrdv 3298 1  |-  ( A 
C_  B  ->  ( C  \  B )  C_  ( C  \  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1717    \ cdif 3261    C_ wss 3264
This theorem is referenced by:  sscond  3428  sorpsscmpl  6470  sbthlem1  7154  sbthlem2  7155  cantnfp1lem1  7568  cantnfp1lem3  7570  isf34lem7  8193  isf34lem6  8194  setsres  13423  mplsubglem  16426  cctop  16994  clsval2  17038  ntrss  17043  hauscmplem  17392  ptbasin  17531  cfinfil  17847  csdfil  17848  uniioombllem5  19347  kur14lem6  24677  dvreasin  25981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-dif 3267  df-in 3271  df-ss 3278
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