Definition df-disj 5074

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 5073	. 2 wff Disj 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1569	. . . . 5 class 𝑦
7	6, 3	wcel 2147	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrmo 3368	. . 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  5075  disjss2  5076  cbvdisj  5083  cbvdisjv  5084  nfdisj1  5087  disjor  5088  disjiun  5094  cbvdisjf  33253  disjss1f  33254  disjxun0  33256  disjorf  33261  disjin  33268  disjin2  33269  disjrdx  33273  ddemeas  34973  disjeq1i  37189  disjeq12dv  37212  cbvdisjvw2  37232  cbvdisjdavw  37265  cbvdisjdavw2  37286  iccpartdisj  48786