Mathbox for Andrew Salmon < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  compeq Structured version   Visualization version   GIF version

Theorem compeq 37664
 Description: Equality between two ways of saying "the complement of 𝐴." (Contributed by Andrew Salmon, 15-Jul-2011.)
Assertion
Ref Expression
compeq (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥𝐴}
Distinct variable group:   𝑥,𝐴

Proof of Theorem compeq
StepHypRef Expression
1 compel 37663 . 2 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
21abbi2i 2725 1 (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ∖ cdif 3537 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator