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Mirrors > Home > ILE Home > Th. List > df-in | GIF version |
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union (A ∪ B) (df-un 2916) and difference (A ∖ B) (df-dif 2914). (Contributed by NM, 29-Apr-1994.) |
Ref | Expression |
---|---|
df-in | ⊢ (A ∩ B) = {x ∣ (x ∈ A ∧ x ∈ B)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class A | |
2 | cB | . . 3 class B | |
3 | 1, 2 | cin 2910 | . 2 class (A ∩ B) |
4 | vx | . . . . . 6 setvar x | |
5 | 4 | cv 1241 | . . . . 5 class x |
6 | 5, 1 | wcel 1390 | . . . 4 wff x ∈ A |
7 | 5, 2 | wcel 1390 | . . . 4 wff x ∈ B |
8 | 6, 7 | wa 97 | . . 3 wff (x ∈ A ∧ x ∈ B) |
9 | 8, 4 | cab 2023 | . 2 class {x ∣ (x ∈ A ∧ x ∈ B)} |
10 | 3, 9 | wceq 1242 | 1 wff (A ∩ B) = {x ∣ (x ∈ A ∧ x ∈ B)} |
Colors of variables: wff set class |
This definition is referenced by: dfin5 2919 dfss2 2928 elin 3120 disj 3262 iinxprg 3722 bdcin 9318 |
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