Binary Cartesian Product in Kuratowski Formalization contained in Power Set of Power Set of Union/Proof 1
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Theorem
Let $S$ and $T$ be sets.
Let $S \times T$ be the Cartesian product of $S$ and $T$ realized as a set of ordered pairs in Kuratowski formalization.
Then $S \times T \subseteq \powerset {\powerset {S \cup T} }$.
Proof
Let $x \in S$ and $y \in T$.
We are to show that $\set {\set x, \set {x, y} } \in \powerset {\powerset {S \cup T} }$.
\(\ds x\) | \(\in\) | \(\ds S \cup T\) | Definition of Set Union | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \set x\) | \(\in\) | \(\ds \powerset{S \cup T}\) | Definition of Power Set | ||||||||||
\(\ds y\) | \(\in\) | \(\ds S \cup T\) | Definition of Set Union | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \set {x, y}\) | \(\in\) | \(\ds \powerset {S \cup T}\) | Definition of Power Set | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \set {\set x, \set {x, y} }\) | \(\in\) | \(\ds \powerset {\powerset {S \cup T} }\) | Definition of Power Set |
$\blacksquare$