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Definition df-disj 4813
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 4812 . 2 wff Disj 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1636 . . . . 5 class 𝑦
76, 3wcel 2156 . . . 4 wff 𝑦𝐵
87, 1, 2wrmo 3099 . . 3 wff ∃*𝑥𝐴 𝑦𝐵
98, 5wal 1635 . 2 wff 𝑦∃*𝑥𝐴 𝑦𝐵
104, 9wb 197 1 wff (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
Colors of variables: wff setvar class
This definition is referenced by:  dfdisj2  4814  disjss2  4815  cbvdisj  4822  nfdisj1  4825  disjor  4826  disjiun  4832  cbvdisjf  29710  disjss1f  29711  disjorf  29717  disjin  29724  disjin2  29725  disjrdx  29729  ddemeas  30624  iccpartdisj  41948
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