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Theorem velcomp 3958
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 3472 . 2 𝑥 ∈ V
2 eldif 3953 . 2 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
31, 2mpbiran 706 1 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wcel 2098  Vcvv 3468  cdif 3940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946
This theorem is referenced by:  rnep  5919  compeq  43756  compab  43758
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