Definition df-int 3969

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 3226. (Contributed by NM, 18-Aug-1993.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-int

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	cint 3968	. 2 class ∩ 𝐴
3		vy	. . . . . . 7 setvar 𝑦
4	3	cv 1401	. . . . . 6 class 𝑦
5	4, 1	wcel 2209	. . . . 5 wff 𝑦 ∈ 𝐴
6		vx	. . . . . 6 setvar 𝑥
7	6, 3	wel 2210	. . . . 5 wff 𝑥 ∈ 𝑦
8	5, 7	wi 4	. . . 4 wff (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)
9	8, 3	wal 1400	. . 3 wff ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)
10	9, 6	cab 2224	. 2 class {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)}
11	2, 10	wceq 1402	1 wff ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)}

This definition is referenced by:  dfint2  3970  elint  3974  int0  3982  dfiin2g  4043  bdcint  16820  bdcriota  16826