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Theorem neleqtrd 2863
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
neleqtrd.1 (𝜑 → ¬ 𝐶𝐴)
neleqtrd.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
neleqtrd (𝜑 → ¬ 𝐶𝐵)

Proof of Theorem neleqtrd
StepHypRef Expression
1 neleqtrd.1 . 2 (𝜑 → ¬ 𝐶𝐴)
2 neleqtrd.2 . . 3 (𝜑𝐴 = 𝐵)
32eleq2d 2827 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
41, 3mtbid 324 1 (𝜑 → ¬ 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-clel 2816
This theorem is referenced by:  neleqtrrd  2864  smoord  8405  r1tskina  10822  ofccat  15008  mreexexlem2d  17688  opptgdim2  28753  acopyeu  28842  dimlssid  33683  dochnel  41395  stoweidlem26  46041  fourierdlem60  46181  fourierdlem61  46182  sge00  46391  sge0sn  46394  sge0split  46424
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