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Theorem velcomp 3935
Description: Characterization of setvar elements of the complement of a class. (Contributed by Andrew Salmon, 15-Jul-2011.)
Assertion
Ref Expression
velcomp (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)

Proof of Theorem velcomp
StepHypRef Expression
1 vex 3484 . 2 𝑥 ∈ V
2 eldif 3930 . 2 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
31, 2mpbiran 708 1 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wcel 2115  Vcvv 3481  cdif 3917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3483  df-dif 3923
This theorem is referenced by:  rnep  5785  compeq  41065  compab  41067
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