Definition df-disj 3737

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 ∈ A B ↔ ∀y∃*x ∈ A y ∈ 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 3736	. 2 wff Disj x ∈ A B
5		vy	. . . . . 6 setvar y
6	5	cv 1241	. . . . 5 class y
7	6, 3	wcel 1390	. . . 4 wff y ∈ B
8	7, 1, 2	wrmo 2303	. . 3 wff ∃*x ∈ A y ∈ B
9	8, 5	wal 1240	. 2 wff ∀y∃*x ∈ A y ∈ B
10	4, 9	wb 98	1 wff (Disj x ∈ A B ↔ ∀y∃*x ∈ A y ∈ B)

This definition is referenced by:  dfdisj2  3738  disjss2  3739  cbvdisj  3746  nfdisj1  3749