Definition df-disj 5071

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 5070	. 2 wff Disj 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1569	. . . . 5 class 𝑦
7	6, 3	wcel 2145	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrmo 3364	. . 3 wff ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	wal 1568	. 2 wff ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
10	4, 9	wb 209	1 wff (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)

This definition is used by:  dfdisj2  5072  disjss2  5073  cbvdisj  5080  cbvdisjv  5081  nfdisj1  5084  disjor  5085  disjiun  5091  cbvdisjf  33044  disjss1f  33045  disjxun0  33047  disjorf  33052  disjin  33059  disjin2  33060  disjrdx  33064  ddemeas  34747  disjeq1i  36812  disjeq12dv  36835  cbvdisjvw2  36855  cbvdisjdavw  36888  cbvdisjdavw2  36909  iccpartdisj  48337