Equality of Ordered Pairs

Theorem

Two ordered pairs are equal if and only if corresponding coordinates are equal:

$\tuple {a, b} = \tuple {c, d} \iff a = c \land b = d$

Proof

Necessary Condition

Let $\tuple {a, b}$ and $\tuple {c, d}$ be ordered pairs such that $\tuple {a, b} = \tuple {c, d}$.

Then $a = c$ and $b = d$.

$\Box$

Sufficient Condition

Let $\tuple {a, b}$ and $\tuple {c, d}$ be ordered pairs.

Let $a = c$ and $b = d$.

Then:

$\tuple {a, b} = \tuple {c, d}$

$\blacksquare$

Also see

$a \ne b \iff \tuple {a, b} \ne \tuple {b, a}$

that is:

$\tuple {a, b} = \tuple {b, a} \iff a = b$