Theorem eldifsnneq 4795

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 4794	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴 ≠ 𝐶)
2	1	neneqd 2946	1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2107   ∖ cdif 3946  {csn 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3952  df-sn 4630

This theorem is referenced by:  mpodifsnif  7523  symgextfv  19286  evlslem3  21643  evlslem1  21645  2sqreultblem  26951  qsdrngi  32609  elzdif0  32960  bj-fvsnun1  36136  evlsbagval  41138  selvvvval  41157  clsk3nimkb  42791  fdmdifeqresdif  47017