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Theorem neldifpr2 32204
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
StepHypRef Expression
1 neirr 2948 . 2 ¬ 𝐵𝐵
2 eldifpr 4660 . . 3 (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐵𝐶𝐵𝐴𝐵𝐵))
32simp3bi 1146 . 2 (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐵𝐵)
41, 3mto 196 1 ¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2105  wne 2939  cdif 3945  {cpr 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-dif 3951  df-un 3953  df-sn 4629  df-pr 4631
This theorem is referenced by: (None)
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