Theorem eldifsnneq 4748

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

Step	Hyp	Ref	Expression
1		eldifsni 4747	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴 ≠ 𝐶)
2	1	neneqd 2938	1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114   ∖ cdif 3899  {csn 4581

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3443  df-dif 3905  df-sn 4582

This theorem is referenced by:  mpodifsnif  7475  symgextfv  19351  evlslem3  22039  evlslem1  22041  2sqreultblem  27419  qsdrngi  33578  elzdif0  34139  fineqvnttrclselem1  35279  bj-fvsnun1  37462  evlsbagval  42879  selvvvval  42895  clsk3nimkb  44348  fdmdifeqresdif  48655