# Equivalence of Definitions of Set Equality

## Theorem

The following definitions of the concept of Set Equality are equivalent:

### Definition 1

$S$ and $T$ are equal if and only if they have the same elements:

$S = T \iff \paren {\forall x: x \in S \iff x \in T}$

### Definition 2

$S$ and $T$ are equal if and only if both:

$S$ is a subset of $T$

and

$T$ is a subset of $S$

## Proof

### Definition 1 implies Definition 2

Let $S = T$ by Definition 1.

Then:

 $\displaystyle S$ $=$ $\displaystyle T$ $\displaystyle \leadsto \ \$ $\displaystyle \left({x \in S}\right.$ $\iff$ $\displaystyle \left.{x \in T}\right)$ Definition of Set Equality $\displaystyle \leadsto \ \$ $\displaystyle \left({x \in S}\right.$ $\implies$ $\displaystyle \left.{x \in T}\right)$ Biconditional Elimination $\displaystyle \leadsto \ \$ $\displaystyle S$ $\subseteq$ $\displaystyle T$ Definition of Subset

Similarly:

 $\displaystyle S$ $=$ $\displaystyle T$ $\displaystyle \leadsto \ \$ $\displaystyle \left({x \in S}\right.$ $\iff$ $\displaystyle \left.{x \in T}\right)$ Definition of Set Equality $\displaystyle \leadsto \ \$ $\displaystyle \left({x \in T}\right.$ $\implies$ $\displaystyle \left.{x \in S}\right)$ Biconditional Elimination $\displaystyle \leadsto \ \$ $\displaystyle T$ $\subseteq$ $\displaystyle S$ Definition of Subset

Thus by the Rule of Conjunction:

$S \subseteq T \land T \subseteq S$

and so $S$ and $T$ are equal by Definition 2.

$\Box$

### Definition 2 implies Definition 1

Let $S = T$ by Definition 2:

$S \subseteq T \land T \subseteq S$

First:

 $\displaystyle S$ $\subseteq$ $\displaystyle T$ $\displaystyle \leadsto \ \$ $\displaystyle \left({x \in S}\right.$ $\implies$ $\displaystyle \left.{x \in T}\right)$ Definition of Subset

Then:

 $\displaystyle T$ $\subseteq$ $\displaystyle S$ $\displaystyle \leadsto \ \$ $\displaystyle \left({x \in T}\right.$ $\implies$ $\displaystyle \left.{x \in S}\right)$ Definition of Subset

Thus by Biconditional Introduction:

$\forall x: \paren {x \in S \iff x \in T}$

and so $S$ and $T$ are equal by Definition 1.

$\blacksquare$