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Theorem neleqtrrd 2532
Description: If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
neleqtrrd.1  |-  ( ph  ->  -.  C  e.  B
)
neleqtrrd.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
neleqtrrd  |-  ( ph  ->  -.  C  e.  A
)

Proof of Theorem neleqtrrd
StepHypRef Expression
1 neleqtrrd.1 . 2  |-  ( ph  ->  -.  C  e.  B
)
2 neleqtrrd.2 . . 3  |-  ( ph  ->  A  =  B )
32eleq2d 2503 . 2  |-  ( ph  ->  ( C  e.  A  <->  C  e.  B ) )
41, 3mtbird 293 1  |-  ( ph  ->  -.  C  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725
This theorem is referenced by:  omopth2  6827  mreexd  13867  mreexmrid  13868  lspindp4  16209  lsppratlem3  16221  lebnumlem1  18986  qqhval2lem  24365  qqhf  24370  frlmlbs  27226  psgnunilem2  27395  fnchoice  27676  stoweidlem34  27759  stoweidlem59  27784  mapdindp2  32519  mapdindp4  32521  mapdh6dN  32537  hdmap1l6d  32612
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-cleq 2429  df-clel 2432
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