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Theorem sscon 3473
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 3334 . . . . 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 3322 . . 3  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
5 eldif 3322 . . 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 3346 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 1725    \ cdif 3309    C_ wss 3312
This theorem is referenced by:  sscond  3476  sorpsscmpl  6525  sbthlem1  7209  sbthlem2  7210  cantnfp1lem1  7626  cantnfp1lem3  7628  isf34lem7  8251  isf34lem6  8252  setsres  13487  mplsubglem  16490  cctop  17062  clsval2  17106  ntrss  17111  hauscmplem  17461  ptbasin  17601  cfinfil  17917  csdfil  17918  uniioombllem5  19471  kur14lem6  24889  dvreasin  26280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326
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