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Theorem eldifsnneq 4796
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 4795 . 2 (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴𝐶)
21neneqd 2943 1 (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  cdif 3960  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-sn 4632
This theorem is referenced by:  mpodifsnif  7548  symgextfv  19451  evlslem3  22122  evlslem1  22124  2sqreultblem  27507  qsdrngi  33503  elzdif0  33943  bj-fvsnun1  37238  evlsbagval  42553  selvvvval  42572  clsk3nimkb  44030  fdmdifeqresdif  48187
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