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Theorem eldifsnneq 4684
 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 4683 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴𝐶)
21neneqd 2956 1 (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111   ∖ cdif 3857  {csn 4525 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-v 3411  df-dif 3863  df-sn 4526 This theorem is referenced by:  mpodifsnif  7266  symgextfv  18618  evlslem3  20848  evlslem1  20850  2sqreultblem  26136  elzdif0  31453  bj-fvsnun1  34976  evlsbagval  39808  mhphf  39818  clsk3nimkb  41144  fdmdifeqresdif  45138
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