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Definition df-in 3776
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 27609). Contrast this operation with union (𝐴𝐵) (df-un 3774) and difference (𝐴𝐵) (df-dif 3772). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4062 and dfin4 4069. For intersection defined in terms of union, see dfin3 4068. (Contributed by NM, 29-Apr-1994.)
Assertion
Ref Expression
df-in (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Detailed syntax breakdown of Definition df-in
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
31, 2cin 3768 . 2 class (𝐴𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1636 . . . . 5 class 𝑥
65, 1wcel 2156 . . . 4 wff 𝑥𝐴
75, 2wcel 2156 . . . 4 wff 𝑥𝐵
86, 7wa 384 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4cab 2792 . 2 class {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
103, 9wceq 1637 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfin5  3777  dfss2  3786  elin  3995  disj  4214  iinxprg  4793  disjex  29726  disjexc  29727  eulerpartlemt  30754  iocinico  38291  csbingVD  39608
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