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Theorem compeq 40649
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 3948 . 2 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
21abbi2i 2950 1 (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1528  wcel 2105  {cab 2796  Vcvv 3492  cdif 3930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-dif 3936
This theorem is referenced by: (None)
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