Theorem complexg 4100

Description: The complement of a set is a set. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
complexg	|- (A e. V -> ∼ A e. _V)

Proof of Theorem complexg

Step	Hyp	Ref	Expression
1		df-compl 3213	. 2 |- ∼ A = (A &ncap A)
2		ninexg 4098	. . 3 |- ((A e. V /\ A e. V) -> (A &ncap A) e. _V)
3	2	anidms 626	. 2 |- (A e. V -> (A &ncap A) e. _V)
4	1, 3	syl5eqel 2437	1 |- (A e. V -> ∼ A e. _V)

Syntax hints:   -> wi 4   e. wcel 1710  _Vcvv 2860   &ncap cnin 3205   ∼ ccompl 3206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079

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 2479  df-v 2862  df-nin 3212  df-compl 3213

This theorem is referenced by:  inexg  4101  unexg  4102  difexg  4103  complex  4105  imakexg  4300  intexg  4320  pwexg  4329  imageexg  5801  epprc  5828  fullfunexg  5860  qsexg  5983  addccan2nclem2  6265  fnfreclem1  6318