 Double inclusion proof

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.