Theorem neldifpr1 32448

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 2940	. 2 ⊢ ¬ 𝐴 ≠ 𝐴
2		eldifpr 4632	. . 3 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 ≠ 𝐴 ∧ 𝐴 ≠ 𝐵))
3	2	simp2bi 1146	. 2 ⊢ (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐴 ≠ 𝐴)
4	1, 3	mto 197	1 ⊢ ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})

Syntax hints:  ¬ wn 3   ∈ wcel 2107   ≠ wne 2931   ∖ cdif 3921  {cpr 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-v 3459  df-dif 3927  df-un 3929  df-sn 4600  df-pr 4602

This theorem is referenced by: (None)