Definition df-disj 5115

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 5114	. 2 wff Disj 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1535	. . . . 5 class 𝑦
7	6, 3	wcel 2105	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrmo 3376	. . 3 wff ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	wal 1534	. 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  5116  disjss2  5117  cbvdisj  5124  cbvdisjv  5125  nfdisj1  5128  disjor  5129  disjiun  5135  cbvdisjf  32590  disjss1f  32591  disjxun0  32593  disjorf  32598  disjin  32605  disjin2  32606  disjrdx  32610  ddemeas  34216  disjeq1i  36173  disjeq12dv  36197  cbvdisjvw2  36217  cbvdisjdavw  36250  cbvdisjdavw2  36271  iccpartdisj  47361