Definition df-disj 5087

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 5086	. 2 wff Disj 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1539	. . . . 5 class 𝑦
7	6, 3	wcel 2108	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrmo 3358	. . 3 wff ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	wal 1538	. 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  5088  disjss2  5089  cbvdisj  5096  cbvdisjv  5097  nfdisj1  5100  disjor  5101  disjiun  5107  cbvdisjf  32498  disjss1f  32499  disjxun0  32501  disjorf  32506  disjin  32513  disjin2  32514  disjrdx  32518  ddemeas  34213  disjeq1i  36156  disjeq12dv  36179  cbvdisjvw2  36199  cbvdisjdavw  36232  cbvdisjdavw2  36253  iccpartdisj  47399