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Theorem ssdif 3484
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 3344 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 549 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  -.  x  e.  C
)  ->  ( x  e.  B  /\  -.  x  e.  C ) ) )
3 eldif 3332 . . 3  |-  ( x  e.  ( A  \  C )  <->  ( x  e.  A  /\  -.  x  e.  C ) )
4 eldif 3332 . . 3  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
52, 3, 43imtr4g 263 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( A 
\  C )  ->  x  e.  ( B  \  C ) ) )
65ssrdv 3356 1  |-  ( A 
C_  B  ->  ( A  \  C )  C_  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    e. wcel 1726    \ cdif 3319    C_ wss 3322
This theorem is referenced by:  ssdifd  3485  php  7294  pssnn  7330  mapfien  7656  fin1a2lem13  8297  axcclem  8342  isercolllem3  12465  dprdres  15591  dpjidcl  15621  ablfac1eulem  15635  lspsnat  16222  lbsextlem2  16236  lbsextlem3  16237  mplmonmul  16532  cnsubdrglem  16754  clscon  17498  2ndcdisj2  17525  kqdisj  17769  nulmbl2  19436  i1f1  19585  itg11  19586  itg1climres  19609  limcresi  19777  dvreslem  19801  dvres2lem  19802  dvaddbr  19829  dvmulbr  19830  lhop  19905  elqaa  20244  imadifxp  24043  xrge00  24213  mblfinlem3  26257  mblfinlem4  26258  ismblfin  26259  cnambfre  26267  divrngidl  26652  mvdco  27379  cntzsdrg  27501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336
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