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Theorem ssneldd 3186
Description: If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ssneld.1  |-  ( ph  ->  A  C_  B )
ssneldd.2  |-  ( ph  ->  -.  C  e.  B
)
Assertion
Ref Expression
ssneldd  |-  ( ph  ->  -.  C  e.  A
)

Proof of Theorem ssneldd
StepHypRef Expression
1 ssneldd.2 . 2  |-  ( ph  ->  -.  C  e.  B
)
2 ssneld.1 . . 3  |-  ( ph  ->  A  C_  B )
32ssneld 3185 . 2  |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
41, 3mpd 13 1  |-  ( ph  ->  -.  C  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 2167    C_ wss 3157
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178
This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170
This theorem is referenced by:  0nelrel  4709  addnqprlemfl  7626  addnqprlemfu  7627  mulnqprlemfl  7642  mulnqprlemfu  7643  cauappcvgprlemladdru  7723  fprodntrivap  11749  fprodssdc  11755
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