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Theorem ssdifss 3628
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 3624 . 2  |-  ( A 
\  C )  C_  A
2 sstr 3505 . 2  |-  ( ( ( A  \  C
)  C_  A  /\  A  C_  B )  -> 
( A  \  C
)  C_  B )
31, 2mpan 670 1  |-  ( A 
C_  B  ->  ( A  \  C )  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \ cdif 3466    C_ wss 3469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-dif 3472  df-in 3476  df-ss 3483
This theorem is referenced by:  ssdifssd  3635  xrsupss  11489  xrinfmss  11490  rpnnen2  13809  lpval  19399  lpdifsn  19403  islp2  19405  lpcls  19624  mblfinlem3  29617  mblfinlem4  29618  voliunnfl  29622  fourierdlem42  31404  fourierdlem102  31464  fourierdlem114  31476  lindslinindimp2lem4  32010  lindslinindsimp2lem5  32011  lindslinindsimp2  32012  lincresunit3  32030  ssdifcl  36651  sssymdifcl  36652
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