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Theorem ssdifss 3130
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 3126 . 2  |-  ( A 
\  C )  C_  A
2 sstr 3033 . 2  |-  ( ( ( A  \  C
)  C_  A  /\  A  C_  B )  -> 
( A  \  C
)  C_  B )
31, 2mpan 415 1  |-  ( A 
C_  B  ->  ( A  \  C )  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \ cdif 2996    C_ wss 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012
This theorem is referenced by:  ssdifssd  3138
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