Proof of equivalence classes constituting a partition.

Question

I am not able to understand how the conclusion $[a]$ is a subset of $[c]$ is arrived in this proof. Pls Help.

Theorem Proof:

Answer 1

Let $A$ and $B$ be two sets. Then $A\subseteq B$ if every elements of $A$ are in $B$.

First the author let $e$ be an arbitrary element in $[a]$.
Consequently, the author shows that $e$ must be in $[c]$.
This means that every element of $[a]$ must be in $[c]$ and this shows that $[a]\subseteq[c]$.

Answer 2

We want to prove $e\in[a]\Rightarrow e\in[c]$

If $e\in[a]$ then we have $a\sim e$, and by symmetry $e\sim a$.

On the other hand we have $d\in[a]\cap[c]$ by hypothesis, so $a\sim d$ and $c\sim d$, and this last one implies $d\sim c$ by symmetry.

Since we have $e\sim a$ and $a\sim d$ then we have $e\sim d$ by transitivity.

Now we have $e\sim d$ and $d\sim c$, so by transitivity again $e\sim c$, and by symmetry $c\sim e$, which means $e\in[c]$ as we wanted to show.