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Theorem neldifpr2 32280
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 2943 . 2 ¬ 𝐵𝐵
2 eldifpr 4655 . . 3 (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐵𝐶𝐵𝐴𝐵𝐵))
32simp3bi 1144 . 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 2098  wne 2934  cdif 3940  {cpr 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-v 3470  df-dif 3946  df-un 3948  df-sn 4624  df-pr 4626
This theorem is referenced by: (None)
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