Definition df-disj 4143

Description: A collection of classes B ( x ) is disjoint when for each element y, it is in B ( x ) for at most one x. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
df-disj	|- (Disj x e. A B <-> A. y E* x e. A y e. B )

Distinct variable groups:   x, y   y, A   y, B

Allowed substitution hints:   A( x)   B( x)

Detailed syntax breakdown of Definition df-disj

Step	Hyp	Ref	Expression
1		vx	. . 3 set x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	wdisj 4142	. 2 wff Disj x e. A B
5		vy	. . . . . 6 set y
6	5	cv 1648	. . . . 5 class y
7	6, 3	wcel 1721	. . . 4 wff y e. B
8	7, 1, 2	wrmo 2669	. . 3 wff E* x e. A y e. B
9	8, 5	wal 1546	. 2 wff A. y E* x e. A y e. B
10	4, 9	wb 177	1 wff (Disj x e. A B <-> A. y E* x e. A y e. B )

This definition is referenced by:  dfdisj2  4144  disjss2  4145  cbvdisj  4152  nfdisj1  4155  disjor  4156  disjiun  4162  cbvdisjf  23968  disjss1f  23969  disjorf  23974  disjin  23980  disjrdx  23984