New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  complexg GIF version

Theorem complexg 4099
 Description: The complement of a set is a set. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
complexg (A V → ∼ A V)

Proof of Theorem complexg
StepHypRef Expression
1 df-compl 3212 . 2 A = (AA)
2 ninexg 4097 . . 3 ((A V A V) → (AA) V)
32anidms 626 . 2 (A V → (AA) V)
41, 3syl5eqel 2437 1 (A V → ∼ A V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  Vcvv 2859   ⩃ cnin 3204   ∼ ccompl 3205 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212 This theorem is referenced by:  inexg  4100  unexg  4101  difexg  4102  complex  4104  imakexg  4299  intexg  4319  pwexg  4328  imageexg  5800  epprc  5827  fullfunexg  5859  qsexg  5982  addccan2nclem2  6264  fnfreclem1  6317
 Copyright terms: Public domain W3C validator