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

Step	Hyp	Ref	Expression
1		eldifsni 4790	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴 ≠ 𝐶)
2	1	neneqd 2945	1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶)

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