Theorem eldifsnneq 4742

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 4741	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴 ≠ 𝐶)
2	1	neneqd 2930	1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ∖ cdif 3900  {csn 4577

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3438  df-dif 3906  df-sn 4578

This theorem is referenced by:  mpodifsnif  7464  symgextfv  19297  evlslem3  21985  evlslem1  21987  2sqreultblem  27357  qsdrngi  33441  elzdif0  33963  fineqvnttrclselem1  35090  bj-fvsnun1  37249  evlsbagval  42559  selvvvval  42578  clsk3nimkb  44033  fdmdifeqresdif  48346