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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 3125) and difference (𝐴 ∖ 𝐵) (df-dif 3123). (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 3120 | . 2 class (𝐴 ∩ 𝐵) |
4 | vx | . . . . . 6 setvar 𝑥 | |
5 | 4 | cv 1347 | . . . . 5 class 𝑥 |
6 | 5, 1 | wcel 2141 | . . . 4 wff 𝑥 ∈ 𝐴 |
7 | 5, 2 | wcel 2141 | . . . 4 wff 𝑥 ∈ 𝐵 |
8 | 6, 7 | wa 103 | . . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) |
9 | 8, 4 | cab 2156 | . 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
10 | 3, 9 | wceq 1348 | 1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
Colors of variables: wff set class |
This definition is referenced by: dfin5 3128 dfss2 3136 elin 3310 disj 3462 iinxprg 3945 bdcin 13858 |
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