View source for The intersection of sets is a subset of each set

==Theorem==
That <math>A\cap B\subset A</math> 

Of course by [[Commutivity of intersection|commutivity of <math>\cap</math>]] we have <math>A\cap B\subset B</math> (as <math>A\cap B=B\cap A</math> and <math>B\cap A\subset B</math> by the statement above)<ref>Alec's (my) own work</ref>

==Proof==
We will show <math>x\in A\cap B\implies x\in A</math> then use the [[Implies and subset relation|implies and subset relation]] to conclude <math>A\cap B\subset A</math>


Suppose <math>x\in A\cap B</math>

Then <math>x\in A</math> and <math>x\in B</math>

QED (we have shown that if <math>x\in A\cap B</math> then <math>x\in A</math>, this is what <math>\implies</math> means)

==See also==
* [[Union of subsets is a subset of the union]]
==References==
<references/>
{{Todo|Tidy up this page!}}

{{Theorem Of|Set Theory}}