Vertical Section preserves Increasing Sequences of Sets

Theorem

Let $X$ and $Y$ be sets.

Let $\sequence {A_n}_{n \mathop \in \N}$ be an increasing sequence in $X \times Y$.

Let $x \in X$.

Then:

$\sequence {\paren {A_n}_x}_{n \mathop \in \N}$ is an increasing sequence.

Proof

Since $\sequence {A_n}_{n \mathop \in \N}$ is increasing, we have:

$A_n \subseteq A_{n + 1}$

for each $n$.

From Vertical Section preserves Subsets, we have:

$\paren {A_n}_x \subseteq \paren {A_{n + 1} }_x$

for each $n$.

So:

$\sequence {\paren {A_n}_x}_{n \mathop \in \N}$ is an increasing sequence.

$\blacksquare$