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Definition df-in 3547
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 26674). Contrast this operation with union (𝐴𝐵) (df-un 3545) and difference (𝐴𝐵) (df-dif 3543). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3822 and dfin4 3826. For intersection defined in terms of union, see dfin3 3825. (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 3539 . 2 class (𝐴𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1474 . . . . 5 class 𝑥
65, 1wcel 1977 . . . 4 wff 𝑥𝐴
75, 2wcel 1977 . . . 4 wff 𝑥𝐵
86, 7wa 383 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4cab 2596 . 2 class {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
103, 9wceq 1475 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfin5  3548  dfss2  3557  elin  3758  disj  3969  iinxprg  4537  disjex  28787  disjexc  28788  eulerpartlemt  29760  iocinico  36816  csbingVD  38142
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