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Metamath Proof Explorer

Definition df-int

Description: Define the intersection of a class. Definition 7.35 of TakeutiZaring p. 44. For example, |^| { { 1 , 3 } , { 1 , 8 } } = { 1 } . Compare this with the intersection of two classes, df-in . (Contributed by NM, 18-Aug-1993)

Ref Expression
Assertion df-int ∩ 𝐴 = { 𝑥 ∣ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦 ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ⊢ 𝐴
1 0 cint ⊢ ∩ 𝐴
2 vx ⊢ 𝑥
3 vy ⊢ 𝑦
4 3 cv ⊢ 𝑦
5 4 0 wcel ⊢ 𝑦 ∈ 𝐴
6 2 cv ⊢ 𝑥
7 6 4 wcel ⊢ 𝑥 ∈ 𝑦
8 5 7 wi ⊢ ( 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦 )
9 8 3 wal ⊢ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦 )
10 9 2 cab ⊢ { 𝑥 ∣ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦 ) }
11 1 10 wceq ⊢ ∩ 𝐴 = { 𝑥 ∣ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦 ) }