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Theorem velcomp 3966
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 3961 . 2 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
31, 2mpbiran 709 1 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wcel 2108  Vcvv 3480  cdif 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954
This theorem is referenced by:  rnep  5937  compeq  44459  compab  44461
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