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Theorem ssneld 3069
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 3066 . 2  |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
32con3d 605 1  |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1465    C_ wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054
This theorem is referenced by:  ssneldd  3070  sumdc  11095  summodclem2a  11118  zsumdc  11121  isumss2  11130  decidin  12931
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