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Theorem ssneld 3310
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 3307 . 2  |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
32con3d 127 1  |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1721    C_ wss 3280
This theorem is referenced by:  ssneldd  3311  kmlem2  7987  hashbclem  11656  mrissmrid  13821  mpfrcl  19892  prodss  25226  onsuct0  26095  dvhdimlem  31927  dvh3dim2  31931  dvh3dim3N  31932  mapdh9a  32273  hdmapval0  32319  hdmap11lem2  32328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-in 3287  df-ss 3294
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