Definition – Equality (of sets).

Definition

Let $A$ and $B$ be sets, and let $x\in{A}$ , $x\in{B}$ . We say that $A$ and $B$ are equal if $[x\in{A}\implies x\in{B}] \wedge [x\in{B}\implies x\in{A}]$ .

Notes

The definition is equivalent to proving that $x\in{A} \Leftrightarrow{x\in{A}}$ . This makes sense. If the sets are equal, then we expect them to have exactly the same elements. Thus, if $A=B$ and $A$ contains the element $5$ , for example, we expect the set $B$ to contain the element $5$ as well, and vice versa. I.e, $5\in{A} \Leftrightarrow{5\in{A}}$ .