Double inclusion proof
= Double inclusion proof
{title2=$A=B\iff(A\subseteq B\text{ and }B\subseteq A)$}
To prove equality of two <sets>, show each <set inclusion> separately. If both inclusions hold, every element belongs to one <set> exactly when it belongs to the other; the <axiom of extensionality> gives equality. Each direction may use different hypotheses. For example, when $x\in A$, the condition $x\notin A\setminus B$ implies $x\in B$; without $x\in A$, that inference would be invalid.