Theorem neldifpr2 32790

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 2969	. 2 ⊢ ¬ 𝐵 ≠ 𝐵
2		eldifpr 4620	. . 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 2145   ≠ wne 2960   ∖ cdif 3904  {cpr 4587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3910  df-un 3912  df-sn 4586  df-pr 4588

This theorem is referenced by: (None)