Theorem nincom 3227

Description: Anti-intersection commutes. (Contributed by SF, 10-Jan-2015.)

Assertion

Ref	Expression
nincom	|- (A &ncap B) = (B &ncap A)

Proof of Theorem nincom

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nancom 1290	. . 3 |- ((x e. A -/\ x e. B) <-> (x e. B -/\ x e. A))
2		vex 2863	. . . 4 |- x e. _V
3	2	elnin 3225	. . 3 |- (x e. (A &ncap B) <-> (x e. A -/\ x e. B))
4	2	elnin 3225	. . 3 |- (x e. (B &ncap A) <-> (x e. B -/\ x e. A))
5	1, 3, 4	3bitr4i 268	. 2 |- (x e. (A &ncap B) <-> x e. (B &ncap A))
6	5	eqriv 2350	1 |- (A &ncap B) = (B &ncap A)

Syntax hints:   -/\ wnan 1287   = wceq 1642   e. wcel 1710   &ncap cnin 3205

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

This theorem is referenced by:  nineq2  3236