Full Idea
To know if A ∈ B, we look at the set A as a single object, and check if it is among B's members. But if we want to know whether A ⊆ B then we must open up set A and check whether its various members are among the members of B.
Gist of Idea
∈ says the whole set is in the other; ⊆ says the members of the subset are in the other
Source
Herbert B. Enderton (Elements of Set Theory [1977], 1:04)
Book Reference
Enderton,Herbert B.: 'Elements of Set Theory' [Posts + Telecoms 2006], p.4
A Reaction
This idea is one of the key ideas to grasp if you are going to get the hang of set theory. John ∈ USA ∈ UN, but John is not a member of the UN, because he isn't a country. See Idea 12337 for a special case.
Related Ideas
Idea 12337 There is 'transivity' iff membership ∈ also means inclusion ⊆ [Badiou]
Idea 13443 ∈ relates across layers, while ⊆ relates within layers [Hart,WD]
Idea 15500 Classes divide into subclasses in many ways, but into members in only one way [Lewis]
Idea 15499 A subclass of a subclass is itself a subclass; a member of a member is not in general a member [Lewis]