Theorem complexg 4100

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

Step	Hyp	Ref	Expression
1		df-compl 3213	. 2 ⊢ ∼ A = (A ⩃ A)
2		ninexg 4098	. . 3 ⊢ ((A ∈ V ∧ A ∈ V) → (A ⩃ A) ∈ V)
3	2	anidms 626	. 2 ⊢ (A ∈ V → (A ⩃ A) ∈ V)
4	1, 3	syl5eqel 2437	1 ⊢ (A ∈ V → ∼ A ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1710  Vcvv 2860   ⩃ 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