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Theorem neldifpr1 30632
Description: The first element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
Assertion
Ref Expression
neldifpr1 ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})

Proof of Theorem neldifpr1
StepHypRef Expression
1 neirr 2952 . 2 ¬ 𝐴𝐴
2 eldifpr 4590 . . 3 (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐴𝐶𝐴𝐴𝐴𝐵))
32simp2bi 1148 . 2 (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐴𝐴)
41, 3mto 200 1 ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2112  wne 2943  cdif 3880  {cpr 4560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ne 2944  df-v 3425  df-dif 3886  df-un 3888  df-sn 4559  df-pr 4561
This theorem is referenced by: (None)
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