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Theorem compel 39060
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 3307 . 2 𝑥 ∈ V
2 eldif 3690 . 2 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
31, 2mpbiran 991 1 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wcel 2103  Vcvv 3304  cdif 3677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-v 3306  df-dif 3683
This theorem is referenced by:  compeq  39061  compab  39064  conss34OLD  39065
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