Grothendieck Universe is Closed under Binary Cartesian Product
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Theorem
Let $\mathbb U$ be a Grothendieck universe.
Let $u, v \in \mathbb U$.
Let $u \times v$ be the binary cartesian product of $u$ and $v$ realized as a set of ordered pairs in Kuratowski formalization.
Then $u \times v \in \mathbb U$.
Proof
From Binary Cartesian Product in Kuratowski Formalization contained in Power Set of Power Set of Union:
- $u \times v \subseteq \powerset {\powerset {u \cup v} }$
Then:
\(\ds u \cup v\) | \(\in\) | \(\ds \mathbb U\) | Grothendieck Universe is Closed under Binary Union | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \powerset {u \cup v}\) | \(\in\) | \(\ds \mathbb U\) | Grothendieck Universe: Axiom (3) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \powerset {\powerset {u \cup v} }\) | \(\in\) | \(\ds \mathbb U\) | Grothendieck Universe: Axiom (3) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds u \times v\) | \(\in\) | \(\ds \mathbb U\) | Grothendieck Universe is Closed under Subset |
$\blacksquare$