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Theorem eldifsnneq 4794
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 4793 . 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 1541  wcel 2106  cdif 3945  {csn 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3951  df-sn 4629
This theorem is referenced by:  mpodifsnif  7522  symgextfv  19285  evlslem3  21642  evlslem1  21644  2sqreultblem  26948  qsdrngi  32604  elzdif0  32955  bj-fvsnun1  36131  evlsbagval  41140  selvvvval  41159  clsk3nimkb  42781  fdmdifeqresdif  47007
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