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Theorem eldifsnneq 4791
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 4790 . 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 2108  cdif 3948  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-sn 4627
This theorem is referenced by:  mpodifsnif  7548  symgextfv  19436  evlslem3  22104  evlslem1  22106  2sqreultblem  27492  qsdrngi  33523  elzdif0  33981  bj-fvsnun1  37256  evlsbagval  42576  selvvvval  42595  clsk3nimkb  44053  fdmdifeqresdif  48258
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