Theorem neldifpr2 30418

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 2960	. 2 ⊢ ¬ 𝐵 ≠ 𝐵
2		eldifpr 4557	. . 3 ⊢ (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐵 ∈ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐵 ≠ 𝐵))
3	2	simp3bi 1144	. 2 ⊢ (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐵 ≠ 𝐵)
4	1, 3	mto 200	1 ⊢ ¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})

Syntax hints:  ¬ wn 3   ∈ wcel 2111   ≠ wne 2951   ∖ cdif 3857  {cpr 4527

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-v 3411  df-dif 3863  df-un 3865  df-sn 4526  df-pr 4528

This theorem is referenced by: (None)