# 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 ${⊢}\bigcap {A}=\left\{{x}|\forall {y}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\to {x}\in {y}\right)\right\}$

### Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ${class}{A}$
1 0 cint ${class}\bigcap {A}$
2 vx ${setvar}{x}$
3 vy ${setvar}{y}$
4 3 cv ${setvar}{y}$
5 4 0 wcel ${wff}{y}\in {A}$
6 2 cv ${setvar}{x}$
7 6 4 wcel ${wff}{x}\in {y}$
8 5 7 wi ${wff}\left({y}\in {A}\to {x}\in {y}\right)$
9 8 3 wal ${wff}\forall {y}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\to {x}\in {y}\right)$
10 9 2 cab ${class}\left\{{x}|\forall {y}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\to {x}\in {y}\right)\right\}$
11 1 10 wceq ${wff}\bigcap {A}=\left\{{x}|\forall {y}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\to {x}\in {y}\right)\right\}$