I am given a proof of the following property:
The only left coset of $H$ which is a subgroup of $G$ is $H$ itself.
The brief proof is based on the following properties:
1) $xH = H \iff x\in h$
2) $xH \cap yH =xH$ or $\varnothing$
I understand the proofs of the two above properties. Now the given proof of the coset/subgroup property goes as follows:
Since $e\in H$, $e\not \in gH$ for all $g \in G$ and $g\not \in H$ by properties 1 and 2.
$\therefore$ no coset of $H$ besides H contains $e$, the unique identity in $G$.
$\therefore H $ is the only coset of $H$ which is a subgroup of $G$.
I don't understand what is going on in the first line, it seems that they're trying to find cosets in which the identity from $H$ will not be present. The second line makes sense, it seems relatively obvious to me, though I assume it is based off of the first line... As for the third line it seems like they are 'magically' jumping to it, though I assume that may be because of my lack of understanding of the first line.
So, what is happening in this proof?