Theorem neldifpr1 32860

Description: The first element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)

Assertion

Ref	Expression
neldifpr1	⊢ ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})

Proof of Theorem neldifpr1

Step	Hyp	Ref	Expression
1		neirr 2967	. 2 ⊢ ¬ 𝐴 ≠ 𝐴
2		eldifpr 4625	. . 3 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 ≠ 𝐴 ∧ 𝐴 ≠ 𝐵))
3	2	simp2bi 1164	. 2 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐴 ≠ 𝐴)
4	1, 3	mto 200	1 ⊢ ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})

Syntax hints:  ¬ wn 3   ∈ wcel 2143   ≠ wne 2958   ∖ cdif 3903  {cpr 4592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3909  df-un 3911  df-sn 4591  df-pr 4593

This theorem is referenced by: (None)