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Theorem compeq 44435
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 velcomp 3977 . 2 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
21eqabi 2874 1 (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1536  wcel 2105  {cab 2711  Vcvv 3477  cdif 3959
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-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-dif 3965
This theorem is referenced by: (None)
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