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Theorem neleqtrd 2720
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 2685 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
41, 3mtbid 314 1 (𝜑 → ¬ 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1481  wcel 1988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-cleq 2613  df-clel 2616
This theorem is referenced by:  neleqtrrd  2721  smoord  7447  r1tskina  9589  ofccat  13689  mreexexlem2d  16286  opptgdim2  25618  dochnel  36501  stoweidlem26  40006  fourierdlem60  40146  fourierdlem61  40147  sge00  40356  sge0sn  40359  sge0split  40389  iundjiunlem  40439
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