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Theorem ssneldd 3196
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 3195 . 2  |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
41, 3mpd 14 1  |-  ( ph  ->  -.  C  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1696    C_ wss 3165
This theorem is referenced by:  mreexmrid  13561  mreexexlem2d  13563  acsfiindd  14296  prodrblem  24152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179
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