Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  neldifpr1 Structured version   Visualization version   GIF version

Theorem neldifpr1 32558
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 2946 . 2 ¬ 𝐴𝐴
2 eldifpr 4662 . . 3 (𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐴𝐶𝐴𝐴𝐴𝐵))
32simp2bi 1145 . 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 2105  wne 2937  cdif 3959  {cpr 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-v 3479  df-dif 3965  df-un 3967  df-sn 4631  df-pr 4633
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator