Equality of Ordered Pairs/Sufficient Condition

Theorem

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

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

Then:

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

Proof

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

Then:

$\set a = \set c$

and:

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

Thus:

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

and so by the Kuratowski formalization:

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

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 3$. Ordered pairs; cartesian product sets: Theorem $3.1$
• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions
• 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $9$: Theorem $1.3$