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Theorem eldifsnneq 4738
Description: An element of a difference with a singleton is not equal to the element of that singleton. Note that 𝐴 ∈ {𝐶} → ¬ 𝐴 = 𝐶) need not hold if 𝐴 is a proper class. (Contributed by BJ, 18-Mar-2023.) (Proof shortened by Steven Nguyen, 1-Jun-2023.)
Assertion
Ref Expression
eldifsnneq (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶)

Proof of Theorem eldifsnneq
StepHypRef Expression
1 eldifsni 4737 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴𝐶)
21neneqd 2945 1 (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2105  cdif 3895  {csn 4573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-v 3443  df-dif 3901  df-sn 4574
This theorem is referenced by:  mpodifsnif  7451  symgextfv  19122  evlslem3  21396  evlslem1  21398  2sqreultblem  26702  elzdif0  32228  bj-fvsnun1  35539  evlsbagval  40543  mhphf  40553  clsk3nimkb  41980  fdmdifeqresdif  46037
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