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Theorem ssdif 3108
Description: Difference law for subsets. (Contributed by NM, 28-May-1998.)
Assertion
Ref Expression
ssdif  |-  ( A 
C_  B  ->  ( A  \  C )  C_  ( B  \  C ) )

Proof of Theorem ssdif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssel 2994 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 329 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  -.  x  e.  C
)  ->  ( x  e.  B  /\  -.  x  e.  C ) ) )
3 eldif 2983 . . 3  |-  ( x  e.  ( A  \  C )  <->  ( x  e.  A  /\  -.  x  e.  C ) )
4 eldif 2983 . . 3  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
52, 3, 43imtr4g 203 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( A 
\  C )  ->  x  e.  ( B  \  C ) ) )
65ssrdv 3006 1  |-  ( A 
C_  B  ->  ( A  \  C )  C_  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    e. wcel 1434    \ cdif 2971    C_ wss 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987
This theorem is referenced by:  ssdifd  3109  phpm  6390
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