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Definition df-in 3951
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 30222). Contrast this operation with union (𝐴𝐵) (df-un 3949) and difference (𝐴𝐵) (df-dif 3947). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4256 and dfin4 4263. For intersection defined in terms of union, see dfin3 4262. (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 3943 . 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 2704 . 2 class {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
103, 9wceq 1534 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfin5  3952  elin  3960  dfss2OLD  3965  ss2abdv  4056  disj  4443  disjOLD  4444  iinxprg  5086  disjex  32367  disjexc  32368  eulerpartlemt  33927  iocinico  42563  csbingVD  44246
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