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Theorem neleqtrrd 2292
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 (𝜑 → ¬ 𝐶𝐵)
neleqtrrd.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
neleqtrrd (𝜑 → ¬ 𝐶𝐴)

Proof of Theorem neleqtrrd
StepHypRef Expression
1 neleqtrrd.1 . 2 (𝜑 → ¬ 𝐶𝐵)
2 neleqtrrd.2 . . 3 (𝜑𝐴 = 𝐵)
32eleq2d 2263 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
41, 3mtbird 674 1 (𝜑 → ¬ 𝐶𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1364  wcel 2164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189
This theorem is referenced by:  tfr1onlemsucaccv  6385  tfrcllemsucaccv  6398  zfz1isolemiso  10897  wrdlndm  10918
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