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Definition df-disj 3746
 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 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Detailed syntax breakdown of Definition df-disj
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 cA . . 3 class 𝐴
3 cB . . 3 class 𝐵
41, 2, 3wdisj 3745 . 2 wff Disj 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1242 . . . . 5 class 𝑦
76, 3wcel 1393 . . . 4 wff 𝑦𝐵
87, 1, 2wrmo 2309 . . 3 wff ∃*𝑥𝐴 𝑦𝐵
98, 5wal 1241 . 2 wff 𝑦∃*𝑥𝐴 𝑦𝐵
104, 9wb 98 1 wff (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
 Colors of variables: wff set class This definition is referenced by:  dfdisj2  3747  disjss2  3748  cbvdisj  3755  nfdisj1  3758
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