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Theorem sscon 3178
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 3059 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21con3d 603 . . . 4  |-  ( A 
C_  B  ->  ( -.  x  e.  B  ->  -.  x  e.  A
) )
32anim2d 333 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  C  /\  -.  x  e.  B
)  ->  ( x  e.  C  /\  -.  x  e.  A ) ) )
4 eldif 3048 . . 3  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
5 eldif 3048 . . 3  |-  ( x  e.  ( C  \  A )  <->  ( x  e.  C  /\  -.  x  e.  A ) )
63, 4, 53imtr4g 204 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( C 
\  B )  ->  x  e.  ( C  \  A ) ) )
76ssrdv 3071 1  |-  ( A 
C_  B  ->  ( C  \  B )  C_  ( C  \  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    e. wcel 1463    \ cdif 3036    C_ wss 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052
This theorem is referenced by:  sscond  3181  sbthlem1  6811  sbthlem2  6812
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