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Definition df-in 3567
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 27170). Contrast this operation with union (𝐴𝐵) (df-un 3565) and difference (𝐴𝐵) (df-dif 3563). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3844 and dfin4 3849. For intersection defined in terms of union, see dfin3 3848. (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 3559 . 2 class (𝐴𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1479 . . . . 5 class 𝑥
65, 1wcel 1987 . . . 4 wff 𝑥𝐴
75, 2wcel 1987 . . . 4 wff 𝑥𝐵
86, 7wa 384 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4cab 2607 . 2 class {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
103, 9wceq 1480 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfin5  3568  dfss2  3577  elin  3780  disj  3995  iinxprg  4574  disjex  29291  disjexc  29292  eulerpartlemt  30256  iocinico  37317  csbingVD  38642
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