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Theorem compeq 39423
 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 39422 . 2 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
21abbi2i 2915 1 (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1653   ∈ wcel 2157  {cab 2785  Vcvv 3385   ∖ cdif 3766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772 This theorem is referenced by: (None)
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