# Equivalence of Definitions of Set Equality

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## Theorem

Then the following definitions of set equality are equivalent:

### Definition 1

Two sets are equal if and only if they have the same elements.

This can be defined rigorously as:

$S = T \iff \left({\forall x: x \in S \iff x \in T}\right)$

where $S$ and $T$ are both sets.

### Definition 2

Let $S$ and $T$ be sets.

Then $S$ and $T$ are equal iff:

$S$ 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:

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

Similarly:

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

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

Then:

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

Thus by Biconditional Introduction:

$\forall x: \left({x \in S \iff x \in T}\right)$

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

$\blacksquare$