Theorem neldifpr1 31808

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 2949	. 2 ⊢ ¬ 𝐴 ≠ 𝐴
2		eldifpr 4660	. . 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 196	1 ⊢ ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})

Syntax hints:  ¬ wn 3   ∈ wcel 2106   ≠ wne 2940   ∖ cdif 3945  {cpr 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3951  df-un 3953  df-sn 4629  df-pr 4631

This theorem is referenced by: (None)