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Theorem ssneldd 3003
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 3002 . 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 1434    C_ wss 2974
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 577  ax-in2 578  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987
This theorem is referenced by:  addnqprlemfl  6811  addnqprlemfu  6812  mulnqprlemfl  6827  mulnqprlemfu  6828  cauappcvgprlemladdru  6908
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