Definition df-disj 3945

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 setvar x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	wdisj 3944	. 2 wff Disj x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1334	. . . . 5 class y
7	6, 3	wcel 2128	. . . 4 wff y e. B
8	7, 1, 2	wrmo 2438	. . 3 wff E* x e. A y e. B
9	8, 5	wal 1333	. 2 wff A. y E* x e. A y e. B
10	4, 9	wb 104	1 wff (Disj x e. A B <-> A. y E* x e. A y e. B )

This definition is referenced by:  dfdisj2  3946  disjss2  3947  cbvdisj  3954  nfdisj1  3957  disjnim  3958  disjiun  3962  disjxp1  6185