Theorem neldifpr1 33204

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 2968	. 2 ⊢ ¬ 𝐴 ≠ 𝐴
2		eldifpr 4619	. . 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})

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2147   ≠ wne 2959   ∖ cdif 4048  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-dif 4050  df-un 4154  df-sn 4589  df-pr 4591

This theorem is used by: (None)