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Theorem compel 38144
Description: Equivalence between two ways of saying "is a member of the complement of 𝐴." (Contributed by Andrew Salmon, 15-Jul-2011.)
Assertion
Ref Expression
compel (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)

Proof of Theorem compel
StepHypRef Expression
1 vex 3189 . 2 𝑥 ∈ V
2 eldif 3566 . 2 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
31, 2mpbiran 952 1 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wcel 1987  Vcvv 3186  cdif 3553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3559
This theorem is referenced by:  compeq  38145  compab  38148  conss34OLD  38149
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