Union of Nest of Injections is Injection/Proof

Proof
From Union of Nest of Mappings is Mapping we have that $\bigcup N$ is a mapping.


 * $\exists x_1, x_2: \tuple {x_1, y} \in \bigcup N \land \tuple {x_2, y} \in \bigcup N$
 * $\exists x_1, x_2: \tuple {x_1, y} \in \bigcup N \land \tuple {x_2, y} \in \bigcup N$

such that:
 * $x_1 \ne x_2$

Then:
 * $\exists f \subseteq \bigcup N: \tuple {x_1, y} \in f$

and:
 * $\exists g \subseteq \bigcup N: \tuple {x_1, y} \in g$

But because $N$ is a nest of mappings, either:
 * $f \subseteq g$

or:
 * $g \subseteq f$

That means either:
 * $\tuple {x_1, y} \in g$

or:
 * $\tuple {x_2, y} \in f$

Hence either $f$ or $g$ has both $\tuple {x_1, y}$ and $\tuple {x_2, y}$ in it.

That is, either $f$ or $g$ is not an injection.

From this contradiction it follows that if:
 * $\exists x_1, x_2: \tuple {x_1, y} \in \bigcup N \land \tuple {x_2, y} \in \bigcup N$

then it has to be the case that:
 * $x_1 = x_2$

It follows that $\bigcup N$ is a an injection.