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Definition df-in 3942
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 28132). Contrast this operation with union (𝐴𝐵) (df-un 3940) and difference (𝐴𝐵) (df-dif 3938). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4236 and dfin4 4243. For intersection defined in terms of union, see dfin3 4242. (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 3934 . 2 class (𝐴𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1527 . . . . 5 class 𝑥
65, 1wcel 2105 . . . 4 wff 𝑥𝐴
75, 2wcel 2105 . . . 4 wff 𝑥𝐵
86, 7wa 396 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4cab 2799 . 2 class {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
103, 9wceq 1528 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfin5  3943  dfss2  3954  elin  4168  disj  4397  iinxprg  5003  disjex  30271  disjexc  30272  eulerpartlemt  31529  iocinico  39698  csbingVD  41098
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