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Definition df-in 3872
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 27892). Contrast this operation with union (𝐴𝐵) (df-un 3870) and difference (𝐴𝐵) (df-dif 3868). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4163 and dfin4 4170. For intersection defined in terms of union, see dfin3 4169. (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 3864 . 2 class (𝐴𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1524 . . . . 5 class 𝑥
65, 1wcel 2083 . . . 4 wff 𝑥𝐴
75, 2wcel 2083 . . . 4 wff 𝑥𝐵
86, 7wa 396 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4cab 2777 . 2 class {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
103, 9wceq 1525 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfin5  3873  dfss2  3883  elin  4096  disj  4319  iinxprg  4916  disjex  30028  disjexc  30029  eulerpartlemt  31242  iocinico  39324  csbingVD  40778
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