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Theorem ssneld 3172
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 3169 . 2  |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
32con3d 632 1  |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 2160    C_ wss 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157
This theorem is referenced by:  ssneldd  3173  sumdc  11407  summodclem2a  11430  zsumdc  11433  isumss2  11442  zproddc  11628  prodssdc  11638  decidin  15035
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