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Theorem neldifpr1 32551
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 2949 . 2 ¬ 𝐴𝐴
2 eldifpr 4658 . . 3 (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐴𝐶𝐴𝐴𝐴𝐵))
32simp2bi 1147 . 2 (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐴𝐴)
41, 3mto 197 1 ¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2108  wne 2940  cdif 3948  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by: (None)
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