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Theorem ssdifss 3307
Description: Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
Assertion
Ref Expression
ssdifss  |-  ( A 
C_  B  ->  ( A  \  C )  C_  B )

Proof of Theorem ssdifss
StepHypRef Expression
1 difss 3303 . 2  |-  ( A 
\  C )  C_  A
2 sstr 3187 . 2  |-  ( ( ( A  \  C
)  C_  A  /\  A  C_  B )  -> 
( A  \  C
)  C_  B )
31, 2mpan 651 1  |-  ( A 
C_  B  ->  ( A  \  C )  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \ cdif 3149    C_ wss 3152
This theorem is referenced by:  ssdifssd  3314  unblem1  7109  xrsupss  10627  xrinfmss  10628  rpnnen2  12504  lpval  16871  lpdifsn  16875  islp2  16877  lpcls  17092  ballotlemfrc  23085  islimrs4  25582  lpss2  26468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166
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