Theorem nincompl 4073

Description: Anti-intersection with complement. (Contributed by SF, 2-Jan-2018.)

Assertion

Ref	Expression
nincompl	|- (A &ncap ∼ A) = _V

Proof of Theorem nincompl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqv 3566	. 2 |- ((A &ncap ∼ A) = _V <-> A.x x e. (A &ncap ∼ A))
2		pm3.24 852	. . 3 |- -. (x e. A /\ -. x e. A)
3		vex 2863	. . . . 5 |- x e. _V
4	3	elnin 3225	. . . 4 |- (x e. (A &ncap ∼ A) <-> (x e. A -/\ x e. ∼ A))
5	3	elcompl 3226	. . . . 5 |- (x e. ∼ A <-> -. x e. A)
6	5	nanbi2i 1299	. . . 4 |- ((x e. A -/\ x e. ∼ A) <-> (x e. A -/\ -. x e. A))
7		df-nan 1288	. . . 4 |- ((x e. A -/\ -. x e. A) <-> -. (x e. A /\ -. x e. A))
8	4, 6, 7	3bitri 262	. . 3 |- (x e. (A &ncap ∼ A) <-> -. (x e. A /\ -. x e. A))
9	2, 8	mpbir 200	. 2 |- x e. (A &ncap ∼ A)
10	1, 9	mpgbir 1550	1 |- (A &ncap ∼ A) = _V

Syntax hints:  -. wn 3   /\ wa 358   -/\ wnan 1287   = wceq 1642   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

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:  incompl  4074  uncompl  4075