Theorem neldifpr2 32463

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 2934	. 2 ⊢ ¬ 𝐵 ≠ 𝐵
2		eldifpr 4622	. . 3 ⊢ (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐵 ∈ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐵 ≠ 𝐵))
3	2	simp3bi 1147	. 2 ⊢ (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐵 ≠ 𝐵)
4	1, 3	mto 197	1 ⊢ ¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})

Syntax hints:  ¬ wn 3   ∈ wcel 2109   ≠ wne 2925   ∖ cdif 3911  {cpr 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3449  df-dif 3917  df-un 3919  df-sn 4590  df-pr 4592

This theorem is referenced by: (None)