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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
StepHypRef Expression
1 neirr 2960 . 2 ¬ 𝐵𝐵
2 eldifpr 4557 . . 3 (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐵𝐶𝐵𝐴𝐵𝐵))
32simp3bi 1144 . 2 (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐵𝐵)
41, 3mto 200 1 ¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})
 Colors of variables: wff setvar class 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)
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