Definition df-disj 5070

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 5069	. 2 wff Disj 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1568	. . . . 5 class 𝑦
7	6, 3	wcel 2144	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrmo 3364	. . 3 wff ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	wal 1567	. 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  5071  disjss2  5072  cbvdisj  5079  cbvdisjv  5080  nfdisj1  5083  disjor  5084  disjiun  5090  cbvdisjf  33207  disjss1f  33208  disjxun0  33210  disjorf  33215  disjin  33222  disjin2  33223  disjrdx  33227  ddemeas  34911  disjeq1i  37023  disjeq12dv  37046  cbvdisjvw2  37066  cbvdisjdavw  37099  cbvdisjdavw2  37120  iccpartdisj  48562