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Theorem ssneldd 3507
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 3506 . 2  |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
41, 3mpd 15 1  |-  ( ph  ->  -.  C  e.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1767    C_ wss 3476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-in 3483  df-ss 3490
This theorem is referenced by:  cantnfp1lem3  8095  cantnfp1lem3OLD  8121  fpwwe2lem13  9016  pwfseqlem3  9034  hashbclem  12461  sumrblem  13489  incexclem  13604  ramub1lem2  14397  mreexmrid  14891  mreexexlem2d  14893  acsfiindd  15657  lbspss  17508  lbsextlem4  17587  lindfrn  18620  fclscmpi  20262  lhop2  22148  lhop  22149  dvcnvrelem1  22150  axlowdimlem17  23934  erdszelem8  28279  prodrblem  28635  fprodntriv  28648  fourierdlem80  31487  osumcllem10N  34761  pexmidlem7N  34772  mapdindp2  36518  mapdindp3  36519  hdmapval3lemN  36637  hdmap11lem1  36641
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