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Definition df-in 3714
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 27585). Contrast this operation with union (𝐴𝐵) (df-un 3712) and difference (𝐴𝐵) (df-dif 3710). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3995 and dfin4 4002. For intersection defined in terms of union, see dfin3 4001. (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 3706 . 2 class (𝐴𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1623 . . . . 5 class 𝑥
65, 1wcel 2131 . . . 4 wff 𝑥𝐴
75, 2wcel 2131 . . . 4 wff 𝑥𝐵
86, 7wa 383 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4cab 2738 . 2 class {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
103, 9wceq 1624 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfin5  3715  dfss2  3724  elin  3931  disj  4152  iinxprg  4745  disjex  29704  disjexc  29705  eulerpartlemt  30734  iocinico  38291  csbingVD  39611
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