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Theorem ssdif 3078
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 2939 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 319 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  -.  x  e.  C
)  ->  ( x  e.  B  /\  -.  x  e.  C ) ) )
3 eldif 2927 . . 3  |-  ( x  e.  ( A  \  C )  <->  ( x  e.  A  /\  -.  x  e.  C ) )
4 eldif 2927 . . 3  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
52, 3, 43imtr4g 194 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( A 
\  C )  ->  x  e.  ( B  \  C ) ) )
65ssrdv 2951 1  |-  ( A 
C_  B  ->  ( A  \  C )  C_  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    e. wcel 1393    \ cdif 2914    C_ wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931
This theorem is referenced by:  ssdifd  3079  phpm  6327
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