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Definition df-in 3546
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 26467). Contrast this operation with union (𝐴𝐵) (df-un 3544) and difference (𝐴𝐵) (df-dif 3542). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3821 and dfin4 3825. For intersection defined in terms of union, see dfin3 3824. (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 3538 . 2 class (𝐴𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1473 . . . . 5 class 𝑥
65, 1wcel 1976 . . . 4 wff 𝑥𝐴
75, 2wcel 1976 . . . 4 wff 𝑥𝐵
86, 7wa 382 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4cab 2595 . 2 class {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
103, 9wceq 1474 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfin5  3547  dfss2  3556  elin  3757  disj  3968  iinxprg  4531  disjex  28580  disjexc  28581  eulerpartlemt  29553  iocinico  36599  csbingVD  37925
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