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Theorem neleqtrrd 2236
 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 2207 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
41, 3mtbird 662 1 (𝜑 → ¬ 𝐶𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331   ∈ wcel 1480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133 This theorem is referenced by:  tfr1onlemsucaccv  6231  tfrcllemsucaccv  6244  zfz1isolemiso  10575
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