Theorem neldifpr2 32821

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

Assertion

Ref	Expression
neldifpr2	⊢ ¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})

Proof of Theorem neldifpr2

Step	Hyp	Ref	Expression
1		neirr 2973	. 2 ⊢ ¬ 𝐵 ≠ 𝐵
2		eldifpr 4629	. . 3 ⊢ (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐵 ∈ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐵 ≠ 𝐵))
3	2	simp3bi 1163	. 2 ⊢ (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐵 ≠ 𝐵)
4	1, 3	mto 200	1 ⊢ ¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})

Syntax hints:  ¬ wn 3   ∈ wcel 2149   ≠ wne 2964   ∖ cdif 3910  {cpr 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-v 3465  df-dif 3916  df-un 3918  df-sn 4595  df-pr 4597

This theorem is referenced by: (None)