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Mirrors > Home > MPE Home > Th. List > df-in | Structured version Visualization version GIF version |
Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 28204). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3941) and difference (𝐴 ∖ 𝐵) (df-dif 3939). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4237 and dfin4 4244. For intersection defined in terms of union, see dfin3 4243. (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 3935 | . 2 class (𝐴 ∩ 𝐵) |
4 | vx | . . . . . 6 setvar 𝑥 | |
5 | 4 | cv 1536 | . . . . 5 class 𝑥 |
6 | 5, 1 | wcel 2114 | . . . 4 wff 𝑥 ∈ 𝐴 |
7 | 5, 2 | wcel 2114 | . . . 4 wff 𝑥 ∈ 𝐵 |
8 | 6, 7 | wa 398 | . . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) |
9 | 8, 4 | cab 2799 | . 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
10 | 3, 9 | wceq 1537 | 1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
Colors of variables: wff setvar class |
This definition is referenced by: dfin5 3944 dfss2 3955 elin 4169 disj 4399 iinxprg 5011 disjex 30342 disjexc 30343 eulerpartlemt 31629 iocinico 39867 csbingVD 41267 |
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