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Metamath Proof Explorer

Definition df-in

Description: Define the intersection of two classes. Definition 5.6 of TakeutiZaring p. 16. For example, ( { 1 , 3 } i^i { 1 , 8 } ) = { 1 } ( ex-in ). Contrast this operation with union ( A u. B ) ( df-un ) and difference ( A \ B ) ( df-dif ). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 and dfin4 . For intersection defined in terms of union, see dfin3 . (Contributed by NM, 29-Apr-1994)

Ref Expression
Assertion df-in ( 𝐴 ∩ 𝐵 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ⊢ 𝐴
1 cB ⊢ 𝐵
2 0 1 cin ⊢ ( 𝐴 ∩ 𝐵 )
3 vx ⊢ 𝑥
4 3 cv ⊢ 𝑥
5 4 0 wcel ⊢ 𝑥 ∈ 𝐴
6 4 1 wcel ⊢ 𝑥 ∈ 𝐵
7 5 6 wa ⊢ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 )
8 7 3 cab ⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) }
9 2 8 wceq ⊢ ( 𝐴 ∩ 𝐵 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) }