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Definition df-in 3958
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 30444). Contrast this operation with union (𝐴𝐵) (df-un 3956) and difference (𝐴𝐵) (df-dif 3954). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4271 and dfin4 4278. For intersection defined in terms of union, see dfin3 4277. (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 3950 . 2 class (𝐴𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1539 . . . . 5 class 𝑥
65, 1wcel 2108 . . . 4 wff 𝑥𝐴
75, 2wcel 2108 . . . 4 wff 𝑥𝐵
86, 7wa 395 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4cab 2714 . 2 class {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
103, 9wceq 1540 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfin5  3959  elin  3967  dfss2  3969  disj  4450  iinxprg  5089  disjex  32605  disjexc  32606  eulerpartlemt  34373  in-ax8  36225  iocinico  43224  csbingVD  44904
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