Definition df-disj 5054

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 5053	. 2 wff Disj 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1541	. . . . 5 class 𝑦
7	6, 3	wcel 2114	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrmo 3342	. . 3 wff ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	wal 1540	. 2 wff ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
10	4, 9	wb 206	1 wff (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)

This definition is referenced by:  dfdisj2  5055  disjss2  5056  cbvdisj  5063  cbvdisjv  5064  nfdisj1  5067  disjor  5068  disjiun  5074  cbvdisjf  32656  disjss1f  32657  disjxun0  32659  disjorf  32664  disjin  32671  disjin2  32672  disjrdx  32676  ddemeas  34396  disjeq1i  36390  disjeq12dv  36413  cbvdisjvw2  36433  cbvdisjdavw  36466  cbvdisjdavw2  36487  iccpartdisj  47909