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Theorem ssneld 3030
Description: If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
ssneld.1  |-  ( ph  ->  A  C_  B )
Assertion
Ref Expression
ssneld  |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )

Proof of Theorem ssneld
StepHypRef Expression
1 ssneld.1 . . 3  |-  ( ph  ->  A  C_  B )
21sseld 3027 . 2  |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
32con3d 597 1  |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1439    C_ wss 3002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3008  df-ss 3015
This theorem is referenced by:  ssneldd  3031  sumdc  10810  isummolem2a  10834  zisum  10837  isumss2  10848  decidin  12001
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