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Definition df-in 3954
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 30234). Contrast this operation with union (𝐴𝐵) (df-un 3952) and difference (𝐴𝐵) (df-dif 3950). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4261 and dfin4 4268. For intersection defined in terms of union, see dfin3 4267. (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 3946 . 2 class (𝐴𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1533 . . . . 5 class 𝑥
65, 1wcel 2099 . . . 4 wff 𝑥𝐴
75, 2wcel 2099 . . . 4 wff 𝑥𝐵
86, 7wa 395 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4cab 2705 . 2 class {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
103, 9wceq 1534 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfin5  3955  elin  3963  ss2abdv  4058  disj  4448  disjOLD  4449  iinxprg  5092  disjex  32381  disjexc  32382  eulerpartlemt  33991  iocinico  42640  csbingVD  44323
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