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Theorem ssdifss 3320
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 3316 . 2  |-  ( A 
\  C )  C_  A
2 sstr 3200 . 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 3162    C_ wss 3165
This theorem is referenced by:  ssdifssd  3327  unblem1  7125  xrsupss  10643  xrinfmss  10644  rpnnen2  12520  lpval  16887  lpdifsn  16891  islp2  16893  lpcls  17108  ballotlemfrc  23101  islimrs4  25685  lpss2  26571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179
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