Grothendieck universe
In set theory, a Grothendieck universe is a set U {\displaystyle U} with the following properties: If x {\displaystyle x} is an element of U {\displaystyle U} and if y {\displaystyle y} is an element of x {\displaystyle x} , then y {\displaystyle y} is also an element of U {\displaystyle U} . ( U {\displaystyle U} is a transitive set.) If x {\displaystyle x} and y {\displaystyle y} are both elements of U {\displaystyle U} , then { x , y } {\displaystyle \{x,y\}} is an element of U {\displaystyle U} .