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Theorem ssneld 3456
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 3453 . 2  |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
32con3d 133 1  |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1758    C_ wss 3426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-in 3433  df-ss 3440
This theorem is referenced by:  ssneldd  3457  kmlem2  8421  hashbclem  12307  mrissmrid  14681  mpfrcl  17711  prodss  27594  onsuct0  28421  ftc1anc  28613  dvhdimlem  35395  dvh3dim2  35399  dvh3dim3N  35400  mapdh9a  35741  hdmapval0  35787  hdmap11lem2  35796
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