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Definition df-in 3614
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 27412). Contrast this operation with union (𝐴𝐵) (df-un 3612) and difference (𝐴𝐵) (df-dif 3610). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3893 and dfin4 3900. For intersection defined in terms of union, see dfin3 3899. (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 3606 . 2 class (𝐴𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1522 . . . . 5 class 𝑥
65, 1wcel 2030 . . . 4 wff 𝑥𝐴
75, 2wcel 2030 . . . 4 wff 𝑥𝐵
86, 7wa 383 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4cab 2637 . 2 class {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
103, 9wceq 1523 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfin5  3615  dfss2  3624  elin  3829  disj  4050  iinxprg  4633  disjex  29531  disjexc  29532  eulerpartlemt  30561  iocinico  38114  csbingVD  39434
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